Recent zbMATH articles in MSC 46https://zbmath.org/atom/cc/462022-11-17T18:59:28.764376ZUnknown authorWerkzeugPrefacehttps://zbmath.org/1496.000712022-11-17T18:59:28.764376ZFrom the text: The present volume contains a selection of papers submitted by the participants of the Eleventh International Conference on Function Spaces held in Zielona Góra (July 6--10, 2015) to the issue entitled ``Theory of functions spaces and its applications''.The exponential map for Hopf algebrashttps://zbmath.org/1496.160332022-11-17T18:59:28.764376Z"Alhamzi, Ghaliah"https://zbmath.org/authors/?q=ai:alhamzi.ghaliah"Beggs, Edwin"https://zbmath.org/authors/?q=ai:beggs.edwin-jAuthors' abstract: We give an analogue of the classical exponential map on Lie groups for Hopf \(\star\)-algebras with differential calculus. The major difference with the classical case is the interpretation of the value of the exponential map, classically an element of the Lie group. We give interpretations as states on the Hopf algebra, elements of a Hilbert \(C^\star\)-bimodule of \(\frac{1}{2}\) densities and elements of the dual Hopf algebra. We give examples for complex valued functions on the groups \(S_3\) and \({\mathbb Z}\), Woronowicz's matrix quantum group \({\mathbb C}_q[SU_2]\) and the Sweedler-Taft algebra.
Reviewer: Salih Çelik (İstanbul)Monadic forgetful functors and (non-)presentability for \(C^\ast\)- and \(W^\ast\)-algebrashttps://zbmath.org/1496.180112022-11-17T18:59:28.764376Z"Chirvasitu, Alexandru"https://zbmath.org/authors/?q=ai:chirvasitu.alexandru"Ko, Joanna"https://zbmath.org/authors/?q=ai:ko.joannaThe original impetus for this paper was [\textit{J. Rosický}, Commun. Algebra 50, No. 1, 268--274 (2022; Zbl 1483.18006)] asking whether the forgetful functors
\[
G:\mathcal{C}_{1}^{\ast}\rightarrow\mathrm{Ban}
\]
and
\[
G_{c}:\mathcal{C}_{c,1}^{\ast}\rightarrow\mathrm{Ban}
\]
are monadic, where \(\mathcal{C}_{1}^{\ast}\)\ and \(\mathcal{C}_{c,1}^{\ast} \)\ are the categories of unital \(C^{\ast}\)- and unitial commutative \(C^{\ast} \)-algebras respectively, while \(\mathrm{Ban}\)\ is the category of Banach spaces and linear maps of norm \(\leq1\)\ as morphisms. This paper gives the affirmative answer to the above question (Theorem 2.4 and Corollary 2.6).
Theorem. The forgetful functors from the category \(\mathcal{C}_{1}^{\ast}\) to the categories of unital Banach *-algebras, unital Banach algebras and Banach spaces are all monadic. The same holds for commutative (\(C^{\ast}\)- and Banach) algebras.
The obvious modification of the previous result goes through for von Neumann or \(W^{\ast}\)-algebras (Theorem 4.11 and Corollary 4.13).
Theorem. The forgetful functors from the category \(\mathcal{W}_{1}^{\ast}\)\ of \(W^{\ast}\)-algebras to the categories of \(C^{\ast}\)-algebras, unital Banach *-algebras, unital Banach algebras and Banach spaces are all monadic.
It has been known for some time that the categories \(\mathcal{C}_{1}^{\ast} \)\ and \(\mathcal{C}_{c,1}^{\ast}\)\ are locally \(\aleph_{1}\)-presentable [\textit{J. Adámek} and \textit{J. Rosický}, Locally presentable and accessible categories. Cambridge: Cambridge University Press (1994; Zbl 0795.18007), Theorem 3.28; \textit{J. W. Pelletier} and \textit{J. Rosický}, Algebra Univers. 30, No. 2, 275--284 (1993; Zbl 0817.46057), Theorem 2.4]. A strong negation of local presentability for \(\mathcal{W}_{1}^{\ast}\)\ is demonstrated (Theorem 4.2 and Proposition 4.10).
Theorem. The only presentable objects in the category \(\mathcal{W}_{1}^{\ast}\)\ of von Neumann algebras are \(\left\{ 0\right\} \)\ and \(\mathbb{C}\).
The following speculation is established (Proposition 3.1, Corollary 3.2, Proposition 3.3 and Corollary 3.4).
Theorem. Let \(A\)\ be a commutative unital \(C^{\ast}\)-algebra and \(\mathcal{M}\)\ the class of isometric \(C^{\ast}\) morphisms.
\begin{itemize}
\item \(A\) is \(\aleph_{0}\)-generated with respect to \(\mathcal{M}\)\ in the (plain or enriched) category \(\mathcal{C}_{c,1}^{\ast}\)\ iff it is finite-dimensional.
\item \(A\) is \(\aleph_{0}\)-generated with respect to \(\mathcal{M}\)\ in the ordinary category \(\mathcal{C}_{1}^{\ast}\)\ iff it has dimension \(\leq1\).
\item \(A\) is \(\aleph_{0}\)-generated with respect to \(\mathcal{M}\)\ in the \(\mathrm{CMet}\)-enriched category \(\mathcal{C}_{1}^{\ast}\)\ iff it is finite-dimensional, where \(\mathrm{CMet}\)\ is the category of complete generalized metric spaces.
\end{itemize}
Reviewer: Hirokazu Nishimura (Tsukuba)Quasi-representations of groups and two-homologyhttps://zbmath.org/1496.190012022-11-17T18:59:28.764376Z"Dadarlat, Marius"https://zbmath.org/authors/?q=ai:dadarlat.mariusTwo unitaries whose commutator has small norm give a quasi-representation of the group \(\mathbb{Z}^2\). It may happen that these unitaries are not close to exactly commuting unitaries. \textit{R. Exel} and \textit{T. Loring} [Proc. Am. Math. Soc. 106, No. 4, 913--915 (1989; Zbl 0677.15003)] showed that the nonvanishing of a certain winding number obstructs this. The group \(\mathbb{Z}^2\) is the fundamental group of a torus. The author [J. Topol. Anal. 4, No. 3, 297--319 (2012; Zbl 1258.46029)] has extended the Exel-Loring formula to quasi-representations of the fundamental group \(\Gamma_g\) of an oriented surface of genus \(g\ge1\). The first result in this article generalises this result further to a discrete group together with a class \(x\) in its second integral homology. Via the assembly map, \(x\) gives rise to a class in the \(K\)-theory of the \(\ell^1\)-Banach algebra of the group, and a quasi-representation to a finite matrix algebra attaches an integer to this. The first main theorem identifies this integer with the winding number of a certain loop. An analogous theorem holds for quasi-representations into a unital \(C^\ast\)-algebra with a finite trace.
The setup above is linked to surface groups as follows: there are \(g\ge1\) and a group homomorphism \(f\colon \Gamma_g \to \Gamma\) so that the given homology class \(x\) is the \(f_*\)-image of a canonical element in \(H^2(\Gamma_g,\mathbb{Z})\). This is a key idea for the proof.
The second main result in the article shows under some assumptions that there is a quasi-homomorphism for which the resulting integer is not zero. Of course, this only makes sense if \(x\) is not torsion. In addition, it is assumed that the group admits a \(\gamma\)-element and is isomorphic to a subgroup of the unitary group of a quasidiagonal \(C^\ast\)-algebra. Both assumptions hold, for instance, if the group in question is amenable.
Reviewer: Ralf Meyer (Göttingen)The \(\text{KO}\)-valued spectral flow for skew-adjoint Fredholm operatorshttps://zbmath.org/1496.190022022-11-17T18:59:28.764376Z"Bourne, Chris"https://zbmath.org/authors/?q=ai:bourne.chris"Carey, Alan L."https://zbmath.org/authors/?q=ai:carey.alan-l"Lesch, Matthias"https://zbmath.org/authors/?q=ai:lesch.matthias"Rennie, Adam"https://zbmath.org/authors/?q=ai:rennie.adamThe classical spectral flow is defined for a path of self-adjoint Fredholm operators with invertible end points and counts the number of eigenvalues that cross zero along the path. This article studies the spectral flow for paths of Fredholm operators on real Hilbert spaces with Clifford algebra symmetries, taking values in the \(\text{KO}\)-theory of a point instead of the integers. The \(K\)-theory for real \(C^*\)-algebras has received new attention recently because of its application in the study of symmetry-protected topological phases. Certain symmetries of physical systems such as time reversal are implemented by anti-unitary operators on a Hilbert space, and replace complex by real \(C^*\)-algebras. The spectral flow has already been used for such applications in certain situations, which creates the motivation to develop the theory more systematically.
The article is based on classical results describing the \(\text{KO}\)-theory of a point using modules over Clifford algebras and spaces of Fredholm operators that satisfy appropriate commutation relations with respect to a Clifford algebra representation. More precisely, the authors use a pair of complex structures \(J_0\) and~\(J_1\) that anticommute with the Clifford generators and that satisfy \(\|J_0 - J_1\| <2\). They define an index for such a pair and give an alternative formula for it. Then they define the spectral flow for a continuous path of skew-adjoint Fredholm operators with invertible end points, anticommuting with Clifford generators. They compare their definition with previous definitions and treat some examples related to topological phases. They carry the definition over to paths of unbounded operators and carefully treat the continuity of paths of such operators.
Given various approaches to defining the spectral flow, an important observation in the article is that all reasonable definitions of it agree. Here ``reasonable'' means that the definition should be homotopy invariant, additive for concatenation of paths, and be normalised suitably. This allows to identify the spectral flow with other constructions, by proving that these have the relevant properties as well. This idea is used to identify the spectral flow with the index of a certain Fredholm operator built out of the path, generalising a formula by Robbin and Salamon. The spectral flow is also identified with a Kasparov product in bivariant \(K\)-theory.
Reviewer: Ralf Meyer (Göttingen)Sofic boundaries of groups and coarse geometry of sofic approximationshttps://zbmath.org/1496.200692022-11-17T18:59:28.764376Z"Alekseev, Vadim"https://zbmath.org/authors/?q=ai:alekseev.vadim"Finn-Sell, Martin"https://zbmath.org/authors/?q=ai:finn-sell.martinSummary: Sofic groups generalise both residually finite and amenable groups, and the concept is central to many important results and conjectures in measured group theory. We introduce a topological notion of a sofic boundary attached to a given sofic approximation of a finitely generated group and use it to prove that coarse properties of the approximation (property A, asymptotic coarse embeddability into Hilbert space, geometric property (T)) imply corresponding analytic properties of the group (amenability, a-T-menability and property (T)), thus generalising ideas and results present in the literature for residually finite groups and their box spaces. Moreover, we generalise coarse rigidity results for box spaces due to Kajal Das, proving that coarsely equivalent sofic approximations of two groups give rise to a uniform measure equivalence between those groups. Along the way, we bring to light a coarse geometric viewpoint on ultralimits of a sequence of finite graphs first exposed by Ján Špakula and Rufus Willett, as well as proving some bridging results concerning measure structures on topological groupoid Morita equivalences that will be of interest to groupoid specialists.Density conditions with stabilizers for lattice orbits of Bergman kernels on bounded symmetric domainshttps://zbmath.org/1496.220052022-11-17T18:59:28.764376Z"Caspers, Martijn"https://zbmath.org/authors/?q=ai:caspers.martijn"van Velthoven, Jordy Timo"https://zbmath.org/authors/?q=ai:van-velthoven.jordy-timoThe present paper studies certain estimates which improve on general density theorems for restricted discrete series through the dependence on the stabilizers, while recovering in part sharp results for \(G=\mathrm{PSU}(1,1)\). More precisely, let \(\pi_\alpha\) be a holomorphic discrete series representation of a connected semisimple Lie group \(G\) with finite center, acting on a weighted Bergman space \(A^2_\alpha (\Omega)\) on a bounded symmetric domain \(\Omega\), of formal dimension \(d_{\pi_\alpha}\). The authors show that if the Bergman kernel \(k _Z^{(\alpha)}\) is a cyclic vector for the restriction \(\pi_\alpha|_{\Gamma}\) to a lattice \(\Gamma\le G\), then \(\mathrm{vol}(G/\Gamma)d_{\pi_\alpha}\le|\Gamma_Z|^{-1}\).
Reviewer: Andreas Arvanitoyeorgos (Patras)Lie groups of \(C^k\)-maps on non-compact manifolds and the fundamental theorem for Lie group-valued mappingshttps://zbmath.org/1496.220122022-11-17T18:59:28.764376Z"Alzaareer, Hamza"https://zbmath.org/authors/?q=ai:alzaareer.hamzaThe article studies the existence of Lie group structures on groups of the form \(C^k(M,K), k \in \mathbb{N}\), where \(M\) is a non-compact smooth manifold (possibly with boundary) and \(K\) is a (possibly infinite-dimensional) Lie group. For \(k=\infty\) these groups are known in the literature as current groups. Here Lie group means locally convex Lie group (a la [\textit{K.-H. Neeb}, Jpn. J. Math. (3) 1, No. 2, 291--468 (2006; Zbl 1161.22012)]), where differentiability is understood in the Bastiani calculus. As a tool, a version of the fundamental theorem for Lie algebra-valued functions is provided. Under some technical conditions (involving the regularity of the target Lie group), the author establishes the existence of Lie group structures on \(C^k(M,K)\). In particular, it is shown that \(C^k (\mathbb{R},K)\) admits a Lie group structure under some conditions on \(K\). These results were a stepping stone for the generalised versions on Lie group structures constructed later in [\textit{H. Glöckner} and \textit{A. Schmeding}, Ann. Global Anal. Geom. 61, No. 2, 359--398 (2022; Zbl 1484.58005)].
Reviewer: Alexander Schmeding (Bodø)Clarke Jacobians, Bouligand Jacobians, and compact connected sets of matriceshttps://zbmath.org/1496.260122022-11-17T18:59:28.764376Z"Bartl, David"https://zbmath.org/authors/?q=ai:bartl.david"Fabian, Marián"https://zbmath.org/authors/?q=ai:fabian.marian-j"Kolář, Jan"https://zbmath.org/authors/?q=ai:kolar.janThis note is dedicated to extending from Clarke Jacobians to Bouligand Jacobians various recent results of the first two named authors. The main statement reveals that every nonempty compact connected set of matrices can be expressed as the Bouligand Jacobian at the origin of a suitable Lipschitzian mapping which is moreover either countably piecewise affine or \(C^\infty\)-smooth outside the neighbourhoods of the origin.
Reviewer: Sorin-Mihai Grad (Paris)Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg interpolation inequalities associated with Coulomb-Sobolev spaceshttps://zbmath.org/1496.260222022-11-17T18:59:28.764376Z"Mallick, Arka"https://zbmath.org/authors/?q=ai:mallick.arka"Hoai-Minh Nguyen"https://zbmath.org/authors/?q=ai:nguyen.hoai-minhSummary: We establish the full range Gagliardo-Nirenberg and the Caffarelli-Kohn-Nirenberg interpolation inequalities associated with Coulomb-Sobolev spaces for the (fractional) derivative \(0 \leq s \leq 1\). As a result, we rediscover known Gagliardo-Nirenberg interpolation type inequalities associated with Coulomb-Sobolev spaces which were previously established in the scale of \(H^s\) with \(0 < s \leq 1\) and extend them for the full range \(W^{s, p}\) with \(0 \leq s \leq 1\) and \(1 < p < + \infty\). Using these newly established weighted inequalities, we derive a new family of one body Hardy-Lieb-Thirring inequalities and use it to establish a new family of many body Hardy-Lieb-Thirring inequalities with a strong repulsive interaction term in \(L^p\) scale.On the maximal extension in the mixed ultradifferentiable weight sequence settinghttps://zbmath.org/1496.260422022-11-17T18:59:28.764376Z"Schindl, Gerhard"https://zbmath.org/authors/?q=ai:schindl.gerhardSummary: For the ultradifferentiable weight sequence setting it is known that the Borel map which assigns to each function the infinite jet of derivatives (at \(0)\) is surjective onto the corresponding weighted sequence class if and only if the sequence is strongly nonquasianalytic for both the Roumieu- and Beurling-type classes. Sequences which are nonquasianalytic but not strongly nonquasianalytic admit a controlled loss of regularity and we determine the maximal sequence for which such a mixed setting is possible for both types, hence get information on the controlled loss of surjectivity in this situation. Moreover, we compare the optimal sequences for both mixed strong nonquasianalyticity conditions arising in the literature.Boundary distance functions of Riemann domains over pre-Hilbert spaceshttps://zbmath.org/1496.320482022-11-17T18:59:28.764376Z"Abe, Makoto"https://zbmath.org/authors/?q=ai:abe.makoto"Honda, Tatsuhiro"https://zbmath.org/authors/?q=ai:honda.tatsuhiro"Shima, Tadashi"https://zbmath.org/authors/?q=ai:shima.tadashiSummary: In the present paper, we generalize subpluriharmonic functions in the sense of Fujita to infinite dimension. We show that, for every open set \(D\) in a complex normed space \(E\) and for the boundary distance function \(d\) of \(D\), the function \(-\ln d\) is subpluriharmonic on \(D\). Moreover, we show that, for every Riemann domain \((D, \pi)\) over a complex pre-Hilbert space \(E\) and for the boundary distance function \(d\) of \((D, \pi)\), the function \(-\ln d\) is locally subpluriharmonic on \(D\).Direct proof of termination of the Kohn algorithm in the real-analytic casehttps://zbmath.org/1496.320572022-11-17T18:59:28.764376Z"Nicoara, Andreea C."https://zbmath.org/authors/?q=ai:nicoara.andreea-cSummary: In 1979 \textit{J.J. Kohn} [Acta Math. 142, 79--122 (1979; Zbl 0395.35069)] gave an indirect argument via the Diederich-Fornæss Theorem showing that finite D'Angelo type implies termination of the Kohn algorithm for a pseudoconvex domain with real-analytic boundary. We give here a direct argument for this same implication using the stratification coming from Catlin's notion of a boundary system as well as algebraic geometry on the ring of real-analytic functions. We also indicate how this argument could be used in order to compute an effective lower bound for the subelliptic gain in the \(\overline{\partial}\)-Neumann problem in terms of the D'Angelo type, the dimension of the space, and the level of forms provided that an effective Łojasiewicz inequality can be proven in the real-analytic case and slightly more information obtained about the behavior of the sheaves of multipliers in the Kohn algorithm.The abstract Cauchy problem in locally convex spaceshttps://zbmath.org/1496.340282022-11-17T18:59:28.764376Z"Kruse, Karsten"https://zbmath.org/authors/?q=ai:kruse.karstenSummary: We derive necessary and sufficient criteria for the uniqueness and existence of solutions of the abstract Cauchy problem in locally convex Hausdorff spaces. Our approach is based on a suitable notion of an asymptotic Laplace transform and extends results of Langenbruch beyond the class of Fréchet spaces.Finsler Trudinger-Moser inequalities on \(\mathbb{R}^2\)https://zbmath.org/1496.350182022-11-17T18:59:28.764376Z"Duy, Nguyen Tuan"https://zbmath.org/authors/?q=ai:nguyen-tuan-duy."Phi, Le Long"https://zbmath.org/authors/?q=ai:phi.le-longSummary: The first aim of this article is to study the sharp singular (two-weight) Trudinger-Moser inequalities with Finsler norms on \(\mathbb{R}^2\). The second goal is to propose a different approach to study a vanishing-concentration-compactness principle for the Trudinger-Moser inequalities and use this to investigate the existence and the nonexistence of the maximizers for the Trudinger-Moser inequalities in the subcritical regions. Finally, by applying our Finsler Trudinger-Moser inequalities to suitable Finsler norms, we derive the sharp affine Trudinger-Moser inequalities which are essentially stronger than the Trudinger-Moser inequalities with standard energy of the gradient.Orlicz-Sobolev inequalities and the Dirichlet problem for infinitely degenerate elliptic operatorshttps://zbmath.org/1496.350202022-11-17T18:59:28.764376Z"Hafeez, Usman"https://zbmath.org/authors/?q=ai:hafeez.usman"Lavier, Théo"https://zbmath.org/authors/?q=ai:lavier.theo"Williams, Lucas"https://zbmath.org/authors/?q=ai:williams.lucas"Korobenko, Lyudmila"https://zbmath.org/authors/?q=ai:korobenko.lyudmilaSummary: We investigate a connection between solvability of the Dirichlet problem for an infinitely degenerate elliptic operator and the validity of an Orlicz-Sobolev inequality in the associated subunit metric space. For subelliptic operators it is known that the classical Sobolev inequality is sufficient and almost necessary for the Dirichlet problem to be solvable with a quantitative bound on the solution [\textit{E. T. Sawyer} and \textit{R. L. Wheeden}, Hölder continuity of weak solutions to subelliptic equations with rough coefficients. Providence, RI: American Mathematical Society (AMS) (2006; Zbl 1096.35031)]. When the degeneracy is of infinite type, a weaker Orlicz-Sobolev inequality seems to be the right substitute [\textit{L. Korobenko} et al., Local boundedness, maximum principles, and continuity of solutions to infinitely degenerate elliptic equations with rough coefficients. Providence, RI: American Mathematical Society (AMS) (2021; Zbl 1494.35001)]. In this paper we investigate this connection further and reduce the gap between necessary and sufficient conditions for solvability of the Dirichlet problem.Limiting Sobolev and Hardy inequalities on stratified homogeneous groupshttps://zbmath.org/1496.350232022-11-17T18:59:28.764376Z"van Schaftingen, Jean"https://zbmath.org/authors/?q=ai:van-schaftingen.jean"Yung, Po-Lam"https://zbmath.org/authors/?q=ai:yung.polamSummary: We give a sufficient condition for limiting Sobolev and Hardy inequalities to hold on stratified homogeneous groups. In the Euclidean case, this condition reduces to the known cancelling necessary and sufficient condition. We obtain in particular endpoint Korn-Sobolev and Korn-Hardy inequalities on stratified homogeneous groups.Weak and viscosity solutions for non-homogeneous fractional equations in Orlicz spaceshttps://zbmath.org/1496.351602022-11-17T18:59:28.764376Z"de Borbón, María Laura"https://zbmath.org/authors/?q=ai:de-borbon.maria-laura"Del Pezzo, Leandro M."https://zbmath.org/authors/?q=ai:del-pezzo.leandro-m"Ochoa, Pablo"https://zbmath.org/authors/?q=ai:ochoa.pablo-dSummary: In this paper, we consider non-homogeneous fractional equations in Orlicz spaces, with a source depending on the spatial variable, the unknown function and its fractional gradient. The latter is adapted to the Orlicz framework. The main contribution of the article is to establish the equivalence between weak and viscosity solutions for such equations.The \(s\)-polyharmonic extension problem and higher-order fractional Laplacianshttps://zbmath.org/1496.354232022-11-17T18:59:28.764376Z"Cora, Gabriele"https://zbmath.org/authors/?q=ai:cora.gabriele"Musina, Roberta"https://zbmath.org/authors/?q=ai:musina.robertaSummary: We provide a detailed description of the relationships between the fractional Laplacian of order \(2 s \in(0, n)\) on \(\mathbb{R}^n\) and the \(s\)-\textit{polyharmonic} extension operator.On the full regularity of the free boundary for minima of Alt-Caffarelli functionals in Orlicz spaceshttps://zbmath.org/1496.354572022-11-17T18:59:28.764376Z"Braga, J. Ederson M."https://zbmath.org/authors/?q=ai:braga.j-ederson-m"Regis, Patrícia R. P."https://zbmath.org/authors/?q=ai:regis.patricia-r-pSummary: In this paper, we discuss two issues about the full regularity of the free boundary for overdetermined Bernoulli-type problems in Orlicz spaces. First, we show that in dimension \(n = 2\) there are no singular points on the free boundary \(F(u) := \partial \{ u > 0 \} \cap \Omega\) of minimizers of the Alt-Caffarelli functional
\[
J_G(u) := \int_{\Omega} (G(\vert \nabla u \vert) + \lambda \chi_{\{ u > 0 \}}) dx
\]
for suitable N-functions \(G\). Next, we prove as a consequence of our main results that there exist a critical dimension \(5 \leq n_0 \leq 7\) and a universal constant \(\varepsilon_0 \in (0,1)\) such that if \(G(t)\) is ``\(\varepsilon_0\)-close'' of \(t^2\), then for \(2 \leq n < n_0\), \(F(u)\) is a real analytic hypersurface.On the cohomological equation of a linear contractionhttps://zbmath.org/1496.370032022-11-17T18:59:28.764376Z"Leclercq, Régis"https://zbmath.org/authors/?q=ai:leclercq.regis"Zeggar, Abdellatif"https://zbmath.org/authors/?q=ai:zeggar.abdellatifSummary: In this paper, we study the discrete cohomological equation of a contracting linear automorphism \(A\) of the Euclidean space \(\mathbb{R}^d\). More precisely, if \(\delta\) is the cobord operator defined on the Fréchet space \(E = C^l (\mathbb{R}^d)\) (\(0 \leq l \leq \infty \)) by: \( \delta(h) = h - h \circ A\), we show that:
\begin{itemize}
\item If \(E = C^0(\mathbb{R}^d)\), the range \(\delta(E)\) of \(\delta\) has infinite codimension and its closure is the hyperplane \(E_0\) consisting of the elements of \(E\) vanishing at 0. Consequently, \(H^1 (A, E)\) is infinite dimensional non Hausdorff topological vector space and then the automorphism \(A\) is not cohomologically \(C^0\)-stable.
\item If \(E = C^l(\mathbb{R}^d)\), with \(1 \leq l \leq \infty\), the space \(\delta(E)\) coincides with the closed hyperplane \(E_0\). Consequently, the cohomology space \(H^1 (A, E)\) is of dimension 1 and the automorphism \(A\) is cohomologically \(C^l\)-stable.
\end{itemize}Ternary biderivations and ternary bihomorphisms in \(C^\ast\)-ternary algebrashttps://zbmath.org/1496.390142022-11-17T18:59:28.764376Z"Lee, Jung Rye"https://zbmath.org/authors/?q=ai:lee.jung-rye"Park, Choonkil"https://zbmath.org/authors/?q=ai:park.choonkil"Rassias, Themistocles M."https://zbmath.org/authors/?q=ai:rassias.themistocles-mSummary: In [Rocky Mt. J. Math. 49, No. 2, 593--607 (2019; Zbl 1417.39078)], the second author et al. introduced the following bi-additive \(s\)-functional inequality
\[
\begin{aligned}
\| f(x&+y, z-w) + f(x-y, z+w) -2f(x,z)+2 f(y, w)\| \\
\quad \le \, &\bigg\|s \left(2f \left(\frac{x+y}{2}, z-w \right) + 2f \left(\frac{x-y}{2}, z+w \right) - 2f(x,z)+ 2 f(y, w) \right) \bigg\|,
\end{aligned}
\tag{1}
\]
where \(s\) is a fixed nonzero complex number with \(|s| < 1\). Using the fixed point method, we prove the Hyers-Ulam stability of ternary biderivations and ternary bihomomorphism in \(C^\ast\)-ternary algebras, associated with the bi-additive \(s\)-functional inequality (1).
For the entire collection see [Zbl 1485.65002].Summability of subsequences of a divergent sequence by regular matrices. II.https://zbmath.org/1496.400022022-11-17T18:59:28.764376Z"Boos, Johann"https://zbmath.org/authors/?q=ai:boos.johannSummary: \textit{C. Stuart} proved in [Rocky Mt. J. Math. 44, No. 1, 289--295 (2014; Zbl 1298.40007), Proposition 7] that the Cesàro matrix C 1 cannot sum almost every subsequence of a bounded divergent sequence. At the end of the paper he remarked `It seems likely that this proposition could be generalized for any regular matrix, but we do not have a proof of this'. In [\textit{J. Boos} and \textit{M. Zeltser}, Rocky Mt. J. Math. 48, No. 2, 413--423 (2018; Zbl 1400.40001), Theorem 3.1] Stuart's conjecture is confirmed, and it is even extended to the more general case of divergent sequences. In this note, we show that [loc. cit., Theorem 3.1] is a special case of Theorem 3.5.5 in [\textit{G. M. Petersen}, Regular matrix transformations. New York etc.: McGraw-Hill Publishing Company Ltd. (1966; Zbl 0159.35401)] by proving that the set of all index sequences with positive density is of the second category. For the proof of that a decisive hint was given to the author by Harry I. Miller a few months before he passed away on 17 December 2018.
For Part I see [\textit{J. Boos} and \textit{M. Zeltser}, Rocky Mt. J. Math. 48, No. 2, 413--423 (2018; Zbl 1400.40001)].A generalization of Orlicz sequence spaces derived by quadruple sequential band matrixhttps://zbmath.org/1496.400042022-11-17T18:59:28.764376Z"Dutta, Salila"https://zbmath.org/authors/?q=ai:dutta.salila"Tripathy, Nilambar"https://zbmath.org/authors/?q=ai:tripathy.nilambarSummary: In this article we have introduced a new Orlicz sequence space \(l_p^\lambda(M,B)\) derived by a quadruple sequential band matrix associated with an Orlicz function and lambda matrix. Further, we have studied some topological properties and inclusion relations of this space.On ideal convergence of triple sequences in intuitionistic fuzzy normed space defined by compact operatorhttps://zbmath.org/1496.400122022-11-17T18:59:28.764376Z"Khan, Vakeel A."https://zbmath.org/authors/?q=ai:khan.vakeel-a"Idrisi, Mohd. Imran"https://zbmath.org/authors/?q=ai:idrisi.mohd-imran"Tuba, Umme"https://zbmath.org/authors/?q=ai:tuba.ummeSummary: The main purpose of this article is to introduce and study some new spaces of \(I\)-convergence of triple sequences in intuitionistic fuzzy normed space defined by compact operator i.e., \(_3 S^I_{(\mu,\nu)}(T)\) and \(_3 S^I_{0(\mu,\nu)}(T)\) and examine some fundamental properties, fuzzy topology and verify inclusion relations lying under these spaces.A new type of paranorm intuitionistic fuzzy Zweier \(I\)-convergent double sequence spaceshttps://zbmath.org/1496.400142022-11-17T18:59:28.764376Z"Khan, Vakeel A."https://zbmath.org/authors/?q=ai:khan.vakeel-a"Yasmeen"https://zbmath.org/authors/?q=ai:yasmeen.saba|yasmeen.k-zeba|yasmeen.uzma|yasmeen.farah|yasmeen.shagufta|yasmeen.adeela|yasmeen.hafsa"Fatima, Hira"https://zbmath.org/authors/?q=ai:fatima.hira"Altaf, Henna"https://zbmath.org/authors/?q=ai:altaf.hennaSummary: In this article we introduce the paranorm type intuitionistic fuzzy Zweier \(I\)-convergent double sequence spaces \(_2\mathcal{Z}^I_{(\mu,\nu)}(p)\) and \(_2\mathcal{Z}^I_{0(\mu,\nu)}(p)\) for \(p=(p_{ij})\) a double sequence of positive real numbers and study the fuzzy topology on these spaces.Solvability of some perturbed sequence spaces equations with operatorshttps://zbmath.org/1496.400172022-11-17T18:59:28.764376Z"de Malafosse, Bruno"https://zbmath.org/authors/?q=ai:de-malafosse.bruno"Fares, Ali"https://zbmath.org/authors/?q=ai:fares.ali"Ayad, Ali"https://zbmath.org/authors/?q=ai:ayad.aliSummary: In this paper, we apply the results stated in [\textit{B. de Malafosse} et al., Filomat 32, No. 14, 5123--130 (2018; Zbl 07552731)] to the solvability of the sequence spaces equations (SSE) \(\mathcal{E}+F_x=F_b\), where \(\mathcal{E},F\) are linear spaces of sequences and \(b,x\) are positive sequences (\(x\) is the unknown). In this way, we solve the (SSE) of the form \((E_a)_{G(\alpha,\beta)}+F_x=F_b\), where \(G(\alpha,\beta)\) is a factorable triangle matrix defined by \([G(\alpha, \beta)]_{nk}=\alpha_n\beta_k\) for \(k\leq n\) and \((E,F)\in\{(\ell_\infty,c),(c_0,\ell_\infty),(c_0,c),(\ell^p,c),(\ell^p,\ell_\infty),(w_0,\ell_\infty)\}\) with \(p\geq 1\). Then we deal with some (SSE) involving the matrices \(C(\lambda)\), \(C_1\) and \(\overline{N}_q\). Finally, we solve the (SSE) with operator of the form \((E_a)_{\Sigma^2}+F_x=F_b\).\(L^p\)-Bernstein inequalities on \(C^2\)-domains and applications to discretizationhttps://zbmath.org/1496.410062022-11-17T18:59:28.764376Z"Dai, Feng"https://zbmath.org/authors/?q=ai:dai.feng"Prymak, Andriy"https://zbmath.org/authors/?q=ai:prymak.andriy-vSummary: We prove a new Bernstein type inequality in \(L^p\) spaces associated with the normal and the tangential derivatives on the boundary of a general compact \(C^2\)-domain. We give two applications: Marcinkiewicz type inequality for discretization of \(L^p\) norms and positive cubature formulas. Both results are optimal in the sense that the number of function samples used has the order of the dimension of the corresponding space of algebraic polynomials.Functions with general monotone Fourier coefficientshttps://zbmath.org/1496.420022022-11-17T18:59:28.764376Z"Belov, Aleksandr S."https://zbmath.org/authors/?q=ai:belov.aleksandr-s"Dyachenko, Mikhail I."https://zbmath.org/authors/?q=ai:dyachenko.mikhail-ivanovich"Tikhonov, Sergei Yu."https://zbmath.org/authors/?q=ai:tikhonov.sergey-yuModulation spaces, multipliers associated with the special affine Fourier transformhttps://zbmath.org/1496.420032022-11-17T18:59:28.764376Z"Biswas, M. H. A."https://zbmath.org/authors/?q=ai:biswas.md-haider-ali|biswas.md-hasan-ali"Feichtinger, H. G."https://zbmath.org/authors/?q=ai:feichtinger.hans-georg"Ramakrishnan, R."https://zbmath.org/authors/?q=ai:ramakrishnan.ram-t-s|ramakrishnan.raghu|ramakrishnan.ramya|ramakrishnan.rajasekhar|ramakrishnan.ravi|ramakrishnan.ramkumar|ramakrishnan.rishiSummary: We study some fundamental properties of the special affine Fourier transform (SAFT) in connection with the Fourier analysis and time-frequency analysis. We introduce the modulation space \(\mathcal{M}^{r, s}_A\) in connection with SAFT and prove that if a bounded linear operator between new modulation spaces commutes with \(A\)-translation, then it is a \(A\)-convolution operator. We also establish Hörmander multiplier theorem and Littlewood-Paley theorem associated with the SAFT.Fourier transform of anisotropic mixed-norm Hardy spaces with applications to Hardy-Littlewood inequalitieshttps://zbmath.org/1496.420142022-11-17T18:59:28.764376Z"Liu, Jun"https://zbmath.org/authors/?q=ai:liu.jun.4"Lu, Yaqian"https://zbmath.org/authors/?q=ai:lu.yaqian"Zhang, Mingdong"https://zbmath.org/authors/?q=ai:zhang.mingdongSummary: Let \(\vec{p}\in(0,1]^n\) be an \(n\)-dimensional vector and \(A\) a dilation. Let \(H_A^{\vec{p}}(\mathbb{R}^n)\) denote the anisotropic mixed-norm Hardy space defined via the radial maximal function. Using the known atomic characterization of \(H_A^{\vec{p}}(\mathbb{R}^n)\) and establishing a uniform estimate for corresponding atoms, the authors prove that the Fourier transform of \(f\in H_A^{\vec{p}}(\mathbb{R}^n)\) coincides with a continuous function \(F\) on \(\mathbb{R}^n\) in the sense of tempered distributions. Moreover, the function \(F\) can be controlled pointwisely by the product of the Hardy space norm of \(f\) and a step function with respect to the transpose matrix of \(A\). As applications, the authors obtain a higher order of convergence for the function \(F\) at the origin, and an analogue of Hardy-Littlewood inequalities in the present setting of \(H_A^{\vec{p}}(\mathbb{R}^n)\).The dual spaces of variable anisotropic Hardy-Lorentz spaces and continuity of a class of linear operatorshttps://zbmath.org/1496.420212022-11-17T18:59:28.764376Z"Wang, Wenhua"https://zbmath.org/authors/?q=ai:wang.wenhua"Wang, Aiting"https://zbmath.org/authors/?q=ai:wang.aitingSummary: In this paper, the authors obtain the continuity of a class of linear operators on variable anisotropic Hardy-Lorentz spaces. In addition, the authors also obtain that the dual space of variable anisotropic Hardy-Lorentz spaces is the anisotropic BMO-type space with variable exponents. This result is still new even when the exponent function \(p(\cdot)\) is \(p\).Endpoint Sobolev boundedness and continuity of multilinear fractional maximal functionshttps://zbmath.org/1496.420242022-11-17T18:59:28.764376Z"Liu, Feng"https://zbmath.org/authors/?q=ai:liu.feng"Wu, Huoxiong"https://zbmath.org/authors/?q=ai:wu.huoxiong"Xue, Qingying"https://zbmath.org/authors/?q=ai:xue.qingying"Yabuta, Kôzô"https://zbmath.org/authors/?q=ai:yabuta.kozoThe centered maximal function is denoted \(\mathcal{M}\) and the uncentered maximal function as \(\widetilde{\mathcal{M}}\); I will give the results for the uncentered maximal functions and let the interested reader look at the paper for results related to the centered maximal function. Those results may be slightly different.
There is much that is understood about derivatives of the Hardy-Littlewood maximal function, for example \(\widetilde{\mathcal{M}}:W^{1,p}(\mathbb{R}^n) \mapsto W^{1,p}(\mathbb{R}^n), 1< p \leq \infty\). Because of non-reflexivity, the situation at \(p = 1\) is more delicate, with the basic question being whether \(f \mapsto \nabla\widetilde{ \mathcal{M}}f\) maps \(W^{1,1}(\mathbb{R}^n)\) to \(L^1(\mathbb{R}^n)\). \textit{J. M. Aldaz} and \textit{J. Pérez Lázaro} [Trans. Am. Math. Soc. 359, No. 5, 2443--2461 (2007; Zbl 1143.42021)] proved that if \(f\) is of bounded variation, \(\widetilde{\mathcal{M}}f\) is absolutely continuous and
\[
\mbox{Var} ( \widetilde{\mathcal{M}f}) \leq \mbox{Var}(f),
\]
and thus that
\[
|| (\widetilde{\mathcal{M}}f)^{\prime} ||_{L^1(\mathbb{R})} \leq || f||_{L^1(\mathbb{R})} , f \in W^{1,1}(\mathbb{R}),
\]
and the constant is sharp.
\textit{E. Carneiro} and \textit{J. Madrid} [Trans. Am. Math. Soc. 369, No. 6, 4063--4092 (2017; Zbl 1370.26022)] proved one-dimensional results for the fractional maximal operator
\[
\widetilde{\mathcal{M}}_{\alpha} f(x) = \sup_{r, s \geq 0, r + s>0} \frac{1}{(r +s)^{1 - \alpha}} \int_{x - r}^{x +s} |f(y)| \, dy
\]
(Note that \( \widetilde{\mathcal{M}}_{0} = \widetilde{\mathcal{M}}\)). The ultimate result was in [\textit{J. Madrid}, Rev. Mat. Iberoam. 35, No. 7, 2151--2168 (2019; Zbl 1429.42021)] who proved that if \(0 < \alpha <1, q = \frac{1}{1 - \alpha}, f \mapsto (\widetilde{\mathcal{M}}_{\alpha} f)^{\prime} \) is continuous from \(W^{1,1}(\mathbb{R})\) to \(L^q(\mathbb{R})\).
One of the authors' results is to extend this to the multilinear case. If \(\vec{f} = (f_1, \ldots, f_m)\), where each \(f_j \in L^1_{\mbox{loc}}(\mathbb{R})\), the uncentered fractional maximal operator is
\[
\widetilde{\mathfrak{M}}_{\alpha} \vec{f}(x) = \sup_{r, s \geq 0, r + s>0} \frac{1}{(r +s)^{m - \alpha}} \prod_{j = 1}^m \int_{x - r}^{x +s} |f_j(y)| \, dy,
\]
and it is known that it sends \(W^{1,p_1}(\mathbb{R}) \times \cdots \times W^{1,p_m}(\mathbb{R})\) to \(W^{1,q}(\mathbb{R})\) provided that \(1 < p_1,\dots, p_m, 0 < \alpha <m, 1/q = 1/p_1 + \cdots + 1/p_m - \alpha\). The question is about the behavior of the endpoint case \(p_1 = p_2 = \cdots = p_m = 1\). Question A is whether the mapping \(\vec{f} \mapsto (\widetilde{\mathcal{\mathcal{M}}}_{\alpha} f)^{\prime} \) bounded and continuous from \(W^{1,1}(\mathbb{R}) \times \cdots \times W^{1,1}(\mathbb{R})\) to \(L^q(\mathbb{R})\) if \(0 < \alpha <m , q = \frac{1}{m - \alpha}\). Results of \textit{F. Liu} and \textit{H. Wu} [Can. Math. Bull. 60, No. 3, 586--603 (2017; Zbl 1372.42015)] and others can be extended to handle \(m \geq 2, 1 \leq \alpha < m\) and in this paper the authors prove the remaining case; if \(0 < \alpha <1\), \(q = \frac{1}{1 - \alpha}\), then \(\vec{f} \mapsto (\widetilde{\mathfrak{M}}_{\alpha} f)^{\prime} \) is bounded and continuous from \(W^{1,1}(\mathbb{R}) \times \cdots \times W^{1,1}(\mathbb{R})\) to \(L^q(\mathbb{R})\).
Reviewer: Raymond Johnson (Columbia)Hardy-Littlewood maximal operator on variable Lebesgue spaces with respect to a probability measurehttps://zbmath.org/1496.420252022-11-17T18:59:28.764376Z"Moreno, Jorge"https://zbmath.org/authors/?q=ai:moreno.jorge"Pineda, Ebner"https://zbmath.org/authors/?q=ai:pineda.ebner"Rodriguez, Luz"https://zbmath.org/authors/?q=ai:rodriguez.luz"Urbina, Wilfredo O."https://zbmath.org/authors/?q=ai:urbina-romero.wilfredo-oIn this paper, the authors established the strong and weak boundedness of Hardy-Littlewood maximal operators on variable Lebesgue spaces \(L^{p(\cdot)}(\mu)\) with respect to a probability Borel measure \(\mu\) for two conditions of regularity on the exponent function \(p(\cdot)\).
To be more precise, let \(\mu\) be a Radon measure and \(f\in L^1_{\mathrm{loc}}(\mathbb{R}^d,\mu)\), The Hardy-Littlewood non-centered maximal function of \(f\), with respect to \(\mu\), is defined by
\[
M_\mu f(x)=\sup_{B\ni x}\fint_{B}|f(x)|\mu(dy),
\]
and the Hardy-Littlewood centered maximal function of \(f\), with respect to \(\mu\), is defined by
\[
M_\mu^c f(x)=\sup_{r>0}\fint_{B(x,r)}|f(y)|\mu(dy),
\]
The authors first prove the boundedness with the condition \(\mathcal{P}^0_\mu(\mathbb{R}^d)\).
Theorem 1. Let \(p(\cdot)\in\mathcal{P}^0_\mu(\mathbb{R}^d)\) be continuous with \(p_- > 1\).
(i) There exists \(c>0\) depending on \(p\) such that
\[
\|M_\mu^c f\|_{p(\cdot),\mu}\leq c\|f\|_{p(\cdot),\mu}.
\]
(ii) If \(M_\mu\) is bounded on \(L^p-(\mathbb{R}^d,\mu)\) then there exists \(c>0\) depending on \(p\) such that
\[
\|M_\mu f\|_{p(\cdot),\mu}\leq c\|f\|_{p(\cdot),\mu}.
\]
Then, they gave the boundedness with the condition \(\mathcal{P}_\mu(\mathbb{R}^d)\).
Theorem 2. Let \(p(\cdot)\in\mathcal{P}_\mu(\mathbb{R}^d)\) with \(p_- > 1\) be such that \(1/{p(\cdot)}\) is continuous. If \(M_\mu\) is bounded on \(L^p-(\mathbb{R}^d,\mu)\) then there exists \(c>0\) depending on \(p\) such that
\[
\|M_\mu^c f\|_{p(\cdot),\mu}\leq c\|f\|_{p(\cdot),\mu}.
\]
Theorem 3. Let \(p(\cdot)\in\mathcal{P}_\mu(\mathbb{R}^d)\) be such that \(1/{p(\cdot)}\) is continuous, then there exists \(C>0\) depending on \(p\) such that
\[
\|t\chi_{\{x\in\mathbb{R}^d:M_\mu^c f(x)>t\}}\|_{p(\cdot),\mu}\leq C\|f\|_{p(\cdot),\mu}
\]
for all \(f\in L^1_{\mathrm{loc}}(\mathbb{R}^d,\mu)\), \(t>0\).
They also extended some properties of this operator to a probability Borel measure \(\mu\), the key to extending these results is using the Besicovitch covering lemma instead of the Calderón Zygmund decomposition.
Reviewer: Qingying Xue (Beijing)A note on the boundedness of iterated commutators of multilinear operatorshttps://zbmath.org/1496.420272022-11-17T18:59:28.764376Z"Wang, Dinghuai"https://zbmath.org/authors/?q=ai:wang.dinghuaiSummary: We show that the symbol function belonging to \(BMO\) space is not a necessary condition for the boundedness of the iterated commutator acting on a product of Lebesgue spaces.On BMO and Carleson measures on Riemannian manifoldshttps://zbmath.org/1496.420292022-11-17T18:59:28.764376Z"Brazke, Denis"https://zbmath.org/authors/?q=ai:brazke.denis"Schikorra, Armin"https://zbmath.org/authors/?q=ai:schikorra.armin"Sire, Yannick"https://zbmath.org/authors/?q=ai:sire.yannickSummary: Let \(\mathcal{M}\) be a Riemannian \(n\)-manifold with a metric such that the manifold is Ahlfors regular. We also assume either non-negative Ricci curvature or the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. We characterize BMO-functions \(u: \mathcal{M}\to\mathbb{R}\) by a Carleson measure condition of their \(\sigma\)-harmonic extension \(U: \mathcal{M}\times (0,\infty)\to\mathbb{R}\). We make crucial use of a \(T(b)\) theorem proved by \textit{S. Hofmann} et al. [\(L^p\)-square function estimates on spaces of homogeneous type and on uniformly rectifiable sets. Providence, RI: American Mathematical Society (AMS) (2016; Zbl 1371.28004)]. As an application, we show that the famous theorem of Coifman-Lions-Meyer-Semmes [\textit{R. Coifman} et al., J. Math. Pures Appl. (9) 72, No. 3, 247--286 (1993; Zbl 0864.42009)] holds in this class of manifolds: Jacobians of \(W^{1,n}\)-maps from \(\mathcal{M}\) to \(\mathbb{R}^n\) can be estimated against BMO-functions, which now follows from the arguments for commutators recently proposed by \textit{E. Lenzmann} and \textit{A. Schikorra} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 193, Article ID 111375, 37 p. (2020; Zbl 1436.35012)] using only harmonic extensions, integration by parts, and trace space characterizations.Weak and strong estimates for rough Hausdorff type operator defined on \(p\)-adic linear spacehttps://zbmath.org/1496.420342022-11-17T18:59:28.764376Z"Volosivets, S. S."https://zbmath.org/authors/?q=ai:volosivets.sergey-sergeevichSummary: For rough Hausdorff type operator defined on \(p\)-adic linear space \(Q^n_p\) and its commutator with symbol from Lipschitz space, we give sufficient conditions of their boundedness from one Lorentz space into another.Computational and theoretical aspects of Romanovski-Bessel polynomials and their applications in spectral approximationshttps://zbmath.org/1496.420372022-11-17T18:59:28.764376Z"Zaky, Mahmoud A."https://zbmath.org/authors/?q=ai:zaky.mahmoud-a"Abo-Gabal, Howayda"https://zbmath.org/authors/?q=ai:abo-gabal.howayda"Hafez, Ramy M."https://zbmath.org/authors/?q=ai:hafez.ramy-mahmoud"Doha, Eid H."https://zbmath.org/authors/?q=ai:doha.eid-hThe paper under review presents the main properties of a finite class of orthogonal polynomials with respect to the inverse gamma distribution over the positive real line called Romanovski-Bessel polynomials. More precisely, it introduces the related Romanovski-Bessel-Gauss-type quadrature formulae and the associated interpolation, discrete transforms, spectral differentiation and integration techniques in the physical and frequency spaces, and basic approximation results for the weighted projection operator in weighted Sobolev space. It also addresses the relationship between such kinds of finite orthogonal polynomials and other classes of finite and infinite orthogonal polynomials.
Reviewer: M. Abdessadek Saib (Tebessa)A note on the \(p\)-adic Kozyrev wavelets basishttps://zbmath.org/1496.420442022-11-17T18:59:28.764376Z"Arroyo-Ortiz, Edilberto"https://zbmath.org/authors/?q=ai:arroyo-ortiz.edilbertoThe theory of \(p\)-adic numbers has received considerable attention in several areas of mathematics, including number theory, algebraic geometry, algebraic topology and analysis, and several others. In this article, the authors have formulated a basis of \(p\)-adic wavelets for Sobolev-type spaces consisting of eigenvectors of certain pseudo-differential operators. The obtained results serve as an extension of the fundamental result due to \textit{S. Albeverio} and \textit{S. V. Kozyrev} [\(p\)-Adic Numbers Ultrametric Anal. Appl. 1, No. 3, 181--189 (2009; Zbl 1187.42030)] concerning the multidimensional basis of \(p\)-adic wavelets.
Reviewer: Azhar Y. Tantary (Srinagar)Integral resolvent for Volterra equations and Favard spaceshttps://zbmath.org/1496.450012022-11-17T18:59:28.764376Z"Fadili, A."https://zbmath.org/authors/?q=ai:fadili.ahmed"Maragh, F."https://zbmath.org/authors/?q=ai:maragh.fouadThe authors study the Volterra integral equation
\[
x(t)=x_0 + \int_0^t a(t-s) Ax(s)\, ds,\quad t\geq 0,
\]
in a Banach space \(X\), where \(a\in L^1_{\text{loc}}(\mathbb R^+)\) and \(A\) is a densely define closed operator in \(X\). The integral resolvent associated with this equation is a strongly continuous family \(R(t)\) of bounded operators such that \(R(t)\) commutes with \(A\) and
\[
R(t)x=a(t)x+ \int_0^t a(t-s)AR(s)x\, ds,
\]
for all \(x\in D(A)\).
The crucial assumption used by the authors is that there exists \(\varepsilon_a>0\) and \(t_a>0\) such that for all \(0< t\leq t_a\) one has
\[
\left | \int_0^t a(t-s)a(s)\, ds\right | \geq \varepsilon_a \int_0^t |a(s)|\, ds. \]
They show that
\[
D(A)= \left \{x\in X\,:\, \lim_{t\to 0+} \frac {R(t)x-a(t)x}{(a*a)(t)}\text{ exists}\right \},
\]
and the limit is \(Ax\).
Furthermore, they show that if in addition the integral resolvent is bounded and \(\int_0^\infty \text{e}^{-\omega t} |a(t)|\, dt <\infty\) for some \(\omega >0\) then the following result on the (frequency and temporal) Favard spaces associated with \((A,a)\) holds:
\[
\left \{x\in X\,:\,\sup_{\lambda > \omega}\left \| \frac 1{\hat a(\lambda)} A\left (\frac 1{\hat a(\lambda)}I-A\right)x\right\|<\infty\right \} \]
\[
= \left \{x\in X\,:\,\sup_{0<t\leq 1}\frac{\|R(t)x-a(t)x\|}{ |(a*a)(t)|}<\infty\right \}.
\]
Reviewer: Gustaf Gripenberg (Aalto)Iterative algorithm and theoretical treatment of existence of solution for \((k, z)\)-Riemann-Liouville fractional integral equationshttps://zbmath.org/1496.450052022-11-17T18:59:28.764376Z"Das, Anupam"https://zbmath.org/authors/?q=ai:das.anupam"Rabbani, Mohsen"https://zbmath.org/authors/?q=ai:rabbani.mohsen"Mohiuddine, S. A."https://zbmath.org/authors/?q=ai:mohiuddine.syed-abdul"Deuri, Bhuban Chandra"https://zbmath.org/authors/?q=ai:deuri.bhuban-chandraAfter an introduction to fractional integral equations involving Riemann-Liouville fractional integrals the authors establish a new Darbo-type fixed point theorem. This allows them to discuss the existence of solutions for certain fractional integral equations. The last part of this work is devoted to the construction of a convergent iterative algorithm based on the modified homotopy perturbation method to find the solutions of given fractional integral equations.
Reviewer: Yogesh Sharma (Sardarpura)On the algebraic dimension of Riesz spaceshttps://zbmath.org/1496.460012022-11-17T18:59:28.764376Z"Baziv, N. M."https://zbmath.org/authors/?q=ai:baziv.n-m"Hrybel, O. B."https://zbmath.org/authors/?q=ai:hrybel.o-bSummary: We prove that the algebraic dimension of an infinite dimensional \(C\)-\(\sigma \)-complete Riesz space (in particular, of a Dedekind \(\sigma \)-complete and a laterally \(\sigma \)-complete Riesz space) with the principal projection property which either has a weak order unit or is not purely atomic, is at least continuum. A similar (incomparable to ours) result for complete metric linear spaces is well known.Maximal probability inequalities in vector latticeshttps://zbmath.org/1496.460022022-11-17T18:59:28.764376Z"Divandar, Mahin Sadat"https://zbmath.org/authors/?q=ai:divandar.mahin-sadat"Sadeghi, Ghadir"https://zbmath.org/authors/?q=ai:sadeghi.ghadirSummary: We generalize some maximal probability inequalities, proven for a class of random variables, to the measure-free setting of Riesz spaces. We prove generalizations of the Kolmogorov inequality, Hájek-Rényi inequality, Lévy's inequality and Etemadi's inequality.Sequentially right-like properties on Banach spaceshttps://zbmath.org/1496.460032022-11-17T18:59:28.764376Z"Alikhani, Morteza"https://zbmath.org/authors/?q=ai:alikhani.mortezaSummary: In this paper, we study first the concept of \(p\)-sequentially Right property, which is \(p\)-version of the sequentially Right property. Also, we introduce a new class of subsets of Banach spaces which is called \(p\)-Right\(^\ast\) set and obtain the relationship between \(p\)-Right subsets and \(p\)-Right\(^\ast\) subsets of dual spaces. Furthermore, for \(1\leq p<q\leq\infty\), we introduce the concepts of properties \((SR)_{p,q}\) and \((SR^\ast)_{p,q}\) in order to find a condition such that every Dunford-Pettis \(q\)-convergent operator is Dunford-Pettis \(p\)-convergent. Finally, we apply these concepts and obtain some characterizations of the \(p\)-Dunford-Pettis relatively compact property of Banach spaces and their dual spaces.Relations between two classes of functionshttps://zbmath.org/1496.460042022-11-17T18:59:28.764376Z"Fathi Mourjani, Javad"https://zbmath.org/authors/?q=ai:fathi-mourjani.javadSummary: Let \(\mathbb{F}\) denote a specific space of the class of was constructed by \textit{H. Khodabakhshian} [Ph.D. Thesis. University of Sistan and Baluchestan (2008), per bibl.]\ as a classes of separable Banach function spaces analogous to the James function spaces. In this note, we prove that \(l_p(\alpha)\) is isomorphic to a complemented subspace of \(\mathbb{F}_{\alpha,p}\), and that \(\mathbb{F}_{\alpha,2}\) is a closed subspace of the Waterman-Shiba space \(\alpha BV^2\).On the class of almost L-weakly and almost M-weakly compact operatorshttps://zbmath.org/1496.460052022-11-17T18:59:28.764376Z"Bouras, Khalid"https://zbmath.org/authors/?q=ai:bouras.khalid"Lhaimer, Driss"https://zbmath.org/authors/?q=ai:lhaimer.driss"Moussa, Mohammed"https://zbmath.org/authors/?q=ai:moussa.mohammedSummary: In this paper, we introduce and study new concepts of almost L-weakly and almost M-weakly compact operators.Hypercontractivity and lower deviation estimates in normed spaceshttps://zbmath.org/1496.460062022-11-17T18:59:28.764376Z"Paouris, Grigoris"https://zbmath.org/authors/?q=ai:paouris.grigoris"Tikhomirov, Konstantin"https://zbmath.org/authors/?q=ai:tikhomirov.konstantin-e"Valettas, Petros"https://zbmath.org/authors/?q=ai:valettas.petrosSummary: We consider the problem of estimating small ball probabilities \(\mathbb{P}\{f(G)\le \delta \mathbb{E}f(G)\}\) for subadditive, positively homogeneous functions \(f\) with respect to the Gaussian measure. We establish estimates that depend on global parameters of the underlying function, which take into account analytic and statistical measures, such as the variance and the \({L^1}\)-norms of its partial derivatives. This leads to dimension-dependent bounds for small ball and lower small deviation estimates for seminorms when the linear structure is appropriately chosen to optimize the aforementioned parameters. Our bounds are best possible up to numerical constants. In all regimes, \(\| G\|_{\infty}=\max_{i\le n} |g_i|\) arises as an extremal case in this study. The proofs exploit the convexity and hypercontractivity properties of the Gaussian measure.Predual of \(M^{p,\alpha}(\mathbb{R}^d)\) spaceshttps://zbmath.org/1496.460072022-11-17T18:59:28.764376Z"Kpata, Berenger Akon"https://zbmath.org/authors/?q=ai:kpata.berenger-akonSummary: The space \( M^{p, \alpha} (\mathbb{R}^d)\) introduced by \textit{I. Fofana} [Afr. Mat., Sér. III 5, 53--76 (1995; Zbl 0885.42005)] is a subspace of the Wiener amalgam space of measures. In this note, we give a characterization of a predual space of this one.Uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaceshttps://zbmath.org/1496.460082022-11-17T18:59:28.764376Z"Albiac, F."https://zbmath.org/authors/?q=ai:albiac.fernando"Ansorena, J. L."https://zbmath.org/authors/?q=ai:ansorena.jose-luisSummary: This paper is devoted to providing a unifying approach to the study of the uniqueness of unconditional bases, up to equivalence and permutation, of infinite direct sums of quasi-Banach spaces. Our new approach to this type of problem permits to show that a wide class of vector-valued sequence spaces have a unique unconditional basis up to a permutation. In particular, solving a problem from \textit{F.~Albiac} and \textit{C.~Leránoz} [J. Math. Anal. Appl. 374, No.~2, 394--401 (2011; Zbl 1210.46005)] we show that if \(X\) is quasi-Banach space with a strongly absolute unconditional basis then the infinite direct sum \(\ell_1 (X)\) has a unique unconditional basis up to a permutation, even without knowing whether \(X\) has a unique unconditional basis or not. Applications to the uniqueness of unconditional structure of infinite direct sums of non-locally convex Orlicz and Lorentz sequence spaces, among other classical spaces, are also obtained as a by-product of our work.Unconditional bases in radial Hilbert spaceshttps://zbmath.org/1496.460092022-11-17T18:59:28.764376Z"Isaev, Konstantin P."https://zbmath.org/authors/?q=ai:isaev.konstantin-petrovich"Yulmukhametov, Rinad S."https://zbmath.org/authors/?q=ai:yulmukhametov.rinad-salavatovichCompletion by perturbationshttps://zbmath.org/1496.460102022-11-17T18:59:28.764376Z"Olevskii, Victor"https://zbmath.org/authors/?q=ai:olevskii.victorSummary: Any non-complete orthonormal system in a Hilbert space can be transformed into a basis by small perturbations.Rigidity of the Pu inequality and quadratic isoperimetric constants of normed spaceshttps://zbmath.org/1496.460112022-11-17T18:59:28.764376Z"Creutz, Paul"https://zbmath.org/authors/?q=ai:creutz.paulThe author furnishes an enhanced bound on the filling areas curves (not closed geodesics) in Banach spaces. He shows rigidity of \textit{P. M. Pu}'s classical systolic inequality [Pac. J. Math. 2, 55--71 (1952; Zbl 0046.39902)] and examines the isoperimetric constants of normed spaces.
Reviewer: Mohammed El Aïdi (Bogotá)A norm inequality in James' space and stability of the fixed point propertyhttps://zbmath.org/1496.460122022-11-17T18:59:28.764376Z"Díaz-García, R."https://zbmath.org/authors/?q=ai:diaz-garcia.r"Jiménez-Melado, A."https://zbmath.org/authors/?q=ai:jimenez-melado.antonioSummary: In this paper we prove a norm inequality in James' space \(J\), and use it to show that the fixed point property for nonexpansive mappings is passed on from \(J\) to those Banach spaces \(X\) whose Banach-Mazur distance to \(J\) satisfies \(d(X,J)<\sqrt{\frac{17+\sqrt{97}}{12}}\).Uniform Kadec-Klee properties of Orlicz-Lorentz sequence spaces equipped with the Orlicz normhttps://zbmath.org/1496.460132022-11-17T18:59:28.764376Z"Wang, Di"https://zbmath.org/authors/?q=ai:wang.di.1|wang.di.4|wang.di.3|wang.di.6|wang.di|wang.di.2|wang.di.5|wang.di.7"Cui, Yunan"https://zbmath.org/authors/?q=ai:cui.yunanSummary: Uniform Kadec-Klee property, which takes an indispensable part in the researches of some mathematics branches, has attracted increasing extensive exploration and discussion. In this paper, necessary and sufficient conditions for uniform Kadec-Klee property in Orlicz-Lorentz sequence space equipped with Orlicz norm are given.Abundance of independent sequences in compact spaces and Boolean algebrashttps://zbmath.org/1496.460142022-11-17T18:59:28.764376Z"Avilés, Antonio"https://zbmath.org/authors/?q=ai:aviles.antonio"Martínez-Cervantes, Gonzalo"https://zbmath.org/authors/?q=ai:martinez-cervantes.gonzalo"Plebanek, Grzegorz"https://zbmath.org/authors/?q=ai:plebanek.grzegorzLet \(\mathcal{C}\) be a class of compact spaces. By \(\mathcal{C}^{\perp}\) is denoted the orthogonal class of \(\mathcal{C}\), whose elements are those compact spaces \(K\) such that every continuous image of \(K\) that belongs to \(\mathcal{C}\) is metrizable. The aim of the paper under review is to study the orthogonal class of several well-known classes of compact spaces (denoted by capital letters).
In particular, the article can be split in three main parts. In the first part the authors deal with centeredness. It is shown that the class of compact spaces satisfying the countable chain condition (CCC) is the orthogonal class of the following classes of compact spaces: uniformly Eberlein, Eberlein, Talagrand and Gul'ko (Proposition~9). Moreover, Martin's axiom MA\(_{\omega_1}\) is equivalent to CCC=CORSON\(^{\perp}\) (Proposition~13).
The second part is devoted to the classes of Radon-Nikodým, weakly Radon-Nykodým and the class of weakly Radon-Nikodým Boolean algebras denoted by RN, WRN and WRN(B), respectively. Among others it is proved that dyadic compact spaces are contained in WRN\(^{\perp}\) (Proposition~20). If MA\(_{\omega_1}\) does not hold, then there exists a nonmetrizable zero-dimensional compact space in CORSON \(\cap\) WRN(B)\(^{\perp}\) (Corollary~27). The class WRN(B)\(^{\perp}\setminus\)WRN\(^{\perp}\) contains zero-dimensional compact spaces (Corollary~32).
In the final section, the class of zero-dimensional compact spaces have been investigated. The class ZERO-DIMENSIONAL\(^{\perp}\) is characterized by compact spaces containing at most countably many different clopens (Lemma~33). It is also proved that it is consistent with Martin's axiom and the negation of the continuum hypothesis that there exists a weakly Radon-Nikodým nonmetrizable compact space in ZERO-DIMENSIONAL\(^{(\perp)}\) (Corollary~39).
The symbol \(\mathcal{C}^{(\perp)}\) denotes the class of those compact spaces \(K\) such that every continuous image of any closed subspace of \(K\) that belongs to \(\mathcal{C}\) is metrizable. In Lemma~7 it is shown that \(\mathcal{C}^{(\perp)}\) coincides with the class of hereditarily \(\mathcal{C}^{\perp}\) compact spaces.
Reviewer: Jacopo Somaglia (Pavia)On the class of disjoint limited completely continuous operatorshttps://zbmath.org/1496.460152022-11-17T18:59:28.764376Z"H'michane, Jawad"https://zbmath.org/authors/?q=ai:hmichane.jawad"Hafidi, Noufissa"https://zbmath.org/authors/?q=ai:hafidi.noufissa"Zraoula, Larbi"https://zbmath.org/authors/?q=ai:zraoula.larbiSummary: We introduce and study new class of sets (almost L-limited sets). Also, we introduce new concept of property in Banach lattice (almost Gelfand-Phillips property) and we characterize this property using almost L-limited sets. On the other hand, we introduce the class of disjoint limited completely continuous operators which is a largest class than that of limited completely continuous operators, we characterize this class of operators and we study some of its properties.On the bounded sets in \(C_c (X)\)https://zbmath.org/1496.460162022-11-17T18:59:28.764376Z"Oubbi, Lahbib"https://zbmath.org/authors/?q=ai:oubbi.lahbibSummary: If \(X\) is Hausdorff topological space and \(C_c (X)\) is the topological algebra obtained by endowing the algebra \(C(X)\) of all continuous functions on \(X\) with the topology \(\tau_c\) of uniform convergence on the compact subsets of \(X\), then the set \(\Delta (\varphi) := \{ g \in C(X) : |g(x)| \leq \varphi (x),\,x \in X\}\) is bounded in \(C_c (X)\), for every non-negative \(\varphi \in C(X)\). In this note we deal with the question whether the collection \(C^+\) of all such sets constitutes a base of bounded sets in \(C_c (X)\). We give instances, where the answer is in the affirmative, and others where even the collection \(S^+\) of the sets \(\Delta (\mu)\), with \(\mu\) upper semi-continuous, fails to constitute such a base. We nevertheless provide situations, including the local compact case, where \(S^+\) is a base of bounded sets in \(C_c (X)\).Absence of local unconditional structure in spaces of smooth functions on two-dimensional torushttps://zbmath.org/1496.460172022-11-17T18:59:28.764376Z"Tselishchev, A."https://zbmath.org/authors/?q=ai:tselishchev.a-s|tselishchev.alexey|tselishchev.antonIt is well known that the space \(C^k(\mathbb{T})\) of \(k\) times continuously differentiable functions on the unit circle is isomorphic to \(C(\mathbb{T})\). A quite different situation happens for the space \(C^k(\mathbb{T}^2)\). Under a certain natural condition, it is well known that \(C^k(\mathbb{T}^2)\) is not isomorphic to a complemented subspace of any \(C(S)\) space. In this paper something stronger is proved: \(C^k(\mathbb{T}^2)\) does not have local unconditional structure. Under suitable conditions, it is also proved that \(C^k(\mathbb{T}^2)\) is not isomorphic to any quotient space of any \(C(S)\).
Reviewer: Daniele Puglisi (Catania)The exact spectral asymptotic of the logarithmic potential on harmonic function spacehttps://zbmath.org/1496.460182022-11-17T18:59:28.764376Z"Vujadinović, Djordjije"https://zbmath.org/authors/?q=ai:vujadinovic.djordjijeSummary: In this paper we consider the product of the harmonic Bergman projection \(P_h:L^2(D)\rightarrow L^2_h(D)\) and the operator of logarithmic potential type defined by \(Lf(z)=-\frac{1}{2\pi}\int_D\ln|z-\xi|f(\xi)dA(\xi)\), where \(D\) is the unit disc in \(\mathbb{C}\). We describe the asymptotic behaviour of the eigenvalues of the operator \((P_hL)^\ast(P_hL)\). More precisely, we prove that
\[
\lim\limits_{n\to+\infty} n^2s_n(P_hL)=\sqrt{\frac{4\pi^2}{3}-1}.
\]Two classes of de Branges spaces that are really onehttps://zbmath.org/1496.460192022-11-17T18:59:28.764376Z"Arov, Damir Z."https://zbmath.org/authors/?q=ai:arov.damir-zyamovich"Dym, Harry"https://zbmath.org/authors/?q=ai:dym.harrySummary: It is well known that if \(J\) is an \(m\times m\) signature matrix and \(U\) is \(J\)-inner with respect to the open upper half-plane \(\mathbb{C}_+\), then the kernel
\[
K_\omega^U(\lambda)=\frac{J-U(\lambda)JU(\omega)^\ast}{-2\pi i(\lambda-\overline{\omega})}
\]
is positive and hence is the reproducing kernel of a reproducing kernel Hilbert space \(\mathcal{H}(U)\) of a space of \(m\times 1\) vector valued functions that are holomorphic in the domain of holomorphy of \(U\).
It seems, however, to be not so well known that this reproducing kernel Hilbert space coincides with the de Branges space \(\mathcal{B}(\mathfrak{E})\) based on an appropriately defined de Branges matrix \(\mathfrak{E}=[E_-\;\; E_+]\) with \(m\times m\) components and reproducing kernel
\[
K_\omega^{\mathfrak{E}}(\lambda)=\frac{E_+(\lambda)E_+(\omega)^\ast-E_-(\lambda)E_-(\omega)^\ast}{-2\pi i(\lambda-\overline{\omega})}.
\]
This connection is significant, because it yields a recipe for the inner product in \(\mathcal{H}(U)\) that is not available from Aronszjan's theorem.
Enroute, a pleasing characterization of a class of finite dimensional de Branges spaces \(\mathcal{B}(\mathfrak{E})\) is developed.Unconditional bases of reproducing kernels for Fock spaces with nonradial weightshttps://zbmath.org/1496.460202022-11-17T18:59:28.764376Z"Isaev, K. P."https://zbmath.org/authors/?q=ai:isaev.konstantin-petrovich"Lutsenko, A. V."https://zbmath.org/authors/?q=ai:lutsenko.anastasiya-vladimirovna"Yulmukhametov, R. S."https://zbmath.org/authors/?q=ai:yulmukhametov.rinad-salavatovichSummary: We prove that the Fock space \(\mathcal{F}_\varphi\) with a nonradial weight \(\varphi\) has an unconditional basis of reproducing kernels if and only if such a basis exists for the Fock space \(\mathcal{F}_\varphi\) with a certain radial weight \(\upsilon\) determined by \(\varphi \).On Nikodým and Rainwater sets for \(ba(\mathcal{R})\) and a problem of {M. Valdivia}https://zbmath.org/1496.460212022-11-17T18:59:28.764376Z"Ferrando, J. C."https://zbmath.org/authors/?q=ai:ferrando.juan-carlos"López-Alfonso, S."https://zbmath.org/authors/?q=ai:lopez-alfonso.salvador"López-Pellicer, M."https://zbmath.org/authors/?q=ai:lopez-pellicer.manuelSummary: If \(\mathcal{R}\) is a ring of subsets of a set \(\Omega\) and \(ba(\mathcal{R})\) is the Banach space of bounded finitely additive measures defined on \(\mathcal{R}\) equipped with the supremum norm, a subfamily \(\Delta\) of \(\mathcal{R}\) is called a \textit{Nikodým set} for \(ba(\mathcal{R})\) if each set \(\{\mu_\alpha:\alpha\in\Lambda\}\) in \(ba(\mathcal{R})\) which is pointwise bounded on \(\Delta\) is norm-bounded in \(ba(\mathcal{R})\). If the whole ring \(\mathcal{R}\) is a Nikodým set, \(\mathcal{R}\) is said to have property \((N)\), which means that \(\mathcal{R}\) satisfies the Nikodým-Grothendieck boundedness theorem. In this paper we find a class of rings with property \((N)\) that fail Grothendieck's property \((G)\) and prove that a ring \(\mathcal{R}\) has property \((G)\) if and only if the set of the evaluations on the sets of \(\mathcal{R}\) is a so-called \textit{Rainwater set} for \(ba(\mathcal{R})\). Recalling that \(\mathcal{R}\) is called a \((wN)\)-ring if each increasing web in \(\mathcal{R}\) contains a strand consisting of Nikodým sets, we also give a partial solution to a question raised by \textit{M. Valdivia} [Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 107, No. 2, 355--372 (2013; Zbl 1290.46019)] by providing a class of rings without property \((G)\) for which the relation \((N)\Leftrightarrow(wN)\) holds.The embedding property of the scaling limit of modulation spaceshttps://zbmath.org/1496.460222022-11-17T18:59:28.764376Z"Chen, Jie"https://zbmath.org/authors/?q=ai:chen.jie.1|chen.jie.3|chen.jie|chen.jie.7|chen.jie.8|chen.jie.2|chen.jie.10|chen.jie.5|chen.jie.4|chen.jie.9|chen.jie.6"Lu, Yufeng"https://zbmath.org/authors/?q=ai:lu.yufeng"Wang, Baoxiang"https://zbmath.org/authors/?q=ai:wang.baoxiangSummary: Modulation spaces \(M_{p , q}^s\) were introduced by Feichtinger in 1983. Later, considering the scaling property of the modulation spaces, Sugimoto and Wang [\textit{M.~Sugimoto} and \textit{B.-X. Wang}, Appl. Comput. Harmon. Anal. 53, 54--94 (2021; Zbl 1468.35187)] defined the scaling limit of the modulation spaces, which contains both the modulation spaces and Bényi and Oh's modulation spaces [\textit{Á.~Bényi} and \textit{T.~Oh}, Appl. Comput. Harmon. Anal. 48, No.~1, 496--507 (2020; Zbl 1440.42101)], and these spaces also have some applications in nonlinear Schrödinger equations. So, it is important to consider the relationship between these new spaces and some classical Banach spaces such as \(L^p\) spaces, Fourier \(L^p\) spaces and Besov-Triebel-Sobolev spaces. We study the embedding properties of the scaling limit of the modulation spaces, including the homogeneous case and nonhomogeneous case.Stable Gabor phase retrieval for multivariate functionshttps://zbmath.org/1496.460232022-11-17T18:59:28.764376Z"Grohs, Philipp"https://zbmath.org/authors/?q=ai:grohs.philipp"Rathmair, Martin"https://zbmath.org/authors/?q=ai:rathmair.martinSummary: In recent work [\textit{P.~Grohs} and \textit{M.~Rathmair}, Commun. Pure Appl. Math. 72, No.~5, 981--1043 (2019; Zbl 1460.94022)]
the instabilities of the Gabor phase retrieval problem, i.e., the problem of reconstructing a function \(f\) from its spectrogram \(|\mathcal{G}f|\), where
\[
\mathcal{G}f(x,y)=\int_{\mathbb{R}^d} f(t) e^{-\pi|t-x|^2} e^{-2\pi i t\cdot y} \, dt, \quad x,y\in \mathbb{R}^d,
\] have been completely classified in terms of the disconnectedness of the spectrogram. These findings, however, were crucially restricted to the one-dimensional case (\(d=1\)) and therefore not relevant for many practical applications.
In the present paper we not only generalize the aforementioned results to the multivariate case but also significantly improve on them. Our new results have comprehensive implications in various applications such as ptychography, a highly popular method in coherent diffraction imaging.Uniform estimates with data from generalized Lebesgue spaces in periodic structureshttps://zbmath.org/1496.460242022-11-17T18:59:28.764376Z"Jang, Yunsoo"https://zbmath.org/authors/?q=ai:jang.yunsooSummary: We study various types of uniform Calderón-Zygmund estimates for weak solutions to elliptic equations in periodic homogenization. A global regularity is obtained with respect to the nonhomogeneous term from weighted Lebesgue spaces, Orlicz spaces, and weighted Orlicz spaces, which are generalized Lebesgue spaces, provided that the coefficients have small BMO seminorms and the domains are \(\delta \)-Reifenberg domains.Quotients, \(\ell_\infty\) and abstract Cesàro spaceshttps://zbmath.org/1496.460252022-11-17T18:59:28.764376Z"Kiwerski, Tomasz"https://zbmath.org/authors/?q=ai:kiwerski.tomasz"Kolwicz, Paweł"https://zbmath.org/authors/?q=ai:kolwicz.pawel"Tomaszewski, Jakub"https://zbmath.org/authors/?q=ai:tomaszewski.jakubThe paper deals with Cesàro spaces, proving some general theorems about lattice isometric copies of \(\ell_\infty\) in these spaces.
Reviewer: Daniele Puglisi (Catania)Revisiting Taibleson's theoremhttps://zbmath.org/1496.460262022-11-17T18:59:28.764376Z"Rafeiro, Humberto"https://zbmath.org/authors/?q=ai:rafeiro.humberto"Restrepo, Joel E."https://zbmath.org/authors/?q=ai:restrepo.joel-estebanSummary: A new characterization of the weighted Taibleson's theorem for generalized Hölder spaces is given via a Hadamard-Liouville type operator (Djrbashian's generalized fractional operator).On the existence and nonexistence of isoperimetric inequalities with different monomial weightshttps://zbmath.org/1496.460272022-11-17T18:59:28.764376Z"Abreu, Emerson"https://zbmath.org/authors/?q=ai:abreu.emerson-a-m"Fernandes, Leandro G. jun."https://zbmath.org/authors/?q=ai:fernandes.leandro-g-junSummary: We consider the monomial weight \(x^A=\vert x_1\vert^{a_1}\cdots \vert x_N\vert^{a_N}\), where \(a_i\) is a nonnegative real number for each \(i\in \{1, \dots, N\}\), and we establish the existence and nonexistence of isoperimetric inequalities with different monomial weights. We study positive minimizers of \(\int_{\partial \Omega}x^A \, d\mathcal{H}^{N-1}(x)\) among all smooth bounded open sets \(\Omega\) in \(\mathbb{R}^N\) with fixed Lebesgue measure and monomial weight \(\int_{\Omega }x^B \, dx\). Besides that, we also establish a weighted perimeter inequality under a new version of Steiner symmetrization.Embedding theorems related to torsional rigidity and principal frequencyhttps://zbmath.org/1496.460282022-11-17T18:59:28.764376Z"Avkhadiev, Farit G."https://zbmath.org/authors/?q=ai:avkhadiev.farit-gabidinovichThe logarithmic Sobolev inequality for a submanifold in Euclidean spacehttps://zbmath.org/1496.460292022-11-17T18:59:28.764376Z"Brendle, Simon"https://zbmath.org/authors/?q=ai:brendle.simon.1Summary: We prove a sharp logarithmic Sobolev inequality that holds for submanifolds in Euclidean space of arbitrary dimension and codimension. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature.Improved Moser-Trudinger-Onofri inequality under constraintshttps://zbmath.org/1496.460302022-11-17T18:59:28.764376Z"Chang, Sun-Yung A."https://zbmath.org/authors/?q=ai:chang.sun-yung-alice"Hang, Fengbo"https://zbmath.org/authors/?q=ai:hang.fengboSummary: A classical result of Aubin states that the constant in the Moser-Trudinger-Onofri inequality on \(\mathbb{S}^2\) can be improved for functions with zero first-order moments of the area element [\textit{T. Aubin}, J. Funct. Anal. 32, 148--174 (1979; Zbl 0411.46019)]. We generalize it to the higher-order moments case. These new inequalities bear similarity to a sequence of Lebedev-Milin-type inequalities on \(\mathbb{S}^1\) coming from the work of [\textit{U. Grenander} and \textit{G. Szegö}, Toeplitz forms and their applications. 2nd ed. New York: Chelsea Publishing Company (1984; Zbl 0611.47018)] on Toeplitz determinants (as pointed out by \textit{H. Widom} [Proc. Am. Math. Soc. 102, No. 3, 773--774 (1988; Zbl 0679.42001)]). We also discuss the related sharp inequality by a perturbation method.Nonlocal trace spaces and extension results for nonlocal calculushttps://zbmath.org/1496.460312022-11-17T18:59:28.764376Z"Du, Qiang"https://zbmath.org/authors/?q=ai:du.qiang"Tian, Xiaochuan"https://zbmath.org/authors/?q=ai:tian.xiaochuan"Wright, Cory"https://zbmath.org/authors/?q=ai:wright.cory-d"Yu, Yue"https://zbmath.org/authors/?q=ai:yu.yueSummary: For a given Lipschitz domain \(\Omega \), it is a classical result that the trace space of \(W^{1 , p}(\Omega)\) is \(W^{1 - 1 / p , p}(\partial \Omega)\), namely any \(W^{1 , p}(\Omega)\) function has a well-defined \(W^{1 - 1 / p , p}(\partial \Omega)\) trace on its codimension-1 boundary \(\partial \Omega\) and any \(W^{1 - 1 / p , p}(\partial \Omega)\) function on \(\partial \Omega\) can be extended to a \(W^{1 , p}(\Omega)\) function. In this work, we study function spaces for nonlocal Dirichlet problems involving integrodifferential operators with a finite range of nonlocal interactions, and provide a characterization of their trace spaces. For these nonlocal Dirichlet problems, the boundary conditions are normally imposed on a region with finite thickness volume which lies outside of the domain. We introduce a function space on the volumetric boundary region that serves as a trace space for these nonlocal problems and study the related extension results. Moreover, we discuss the consistency of the new nonlocal trace space with the classical \(W^{1 - 1 / p , p}(\partial \Omega)\) space as the size of nonlocal interaction tends to zero. In making this connection, we conduct an investigation on the relations between nonlocal interactions on a larger domain and the induced interactions on its subdomain. The various forms of trace, embedding and extension theorems may then be viewed as consequences in different scaling limits.Functional calculus on BMO-type spaces of Bourgain, Brezis and Mironescuhttps://zbmath.org/1496.460322022-11-17T18:59:28.764376Z"Liu, Liguang"https://zbmath.org/authors/?q=ai:liu.liguang"Yang, Dachun"https://zbmath.org/authors/?q=ai:yang.dachun"Yuan, Wen"https://zbmath.org/authors/?q=ai:yuan.wenSummary: A nonlinear superposition operator \(T_g\) related to a Borel measurable function \(g: \mathbb{C}\to\mathbb{C}\) is defined via \(T_g(f):=g\circ f\) for any complex-valued function \(f\) on \(\mathbb{R}^n\). This article is devoted to investigating the mapping properties of \(T_g\) on a new BMO type space recently introduced by Bourgain, Brezis and Mironescu [\textit{J.~Bourgain} et al., J. Eur. Math. Soc. (JEMS) 17, No.~9, 2083--2101 (2015; Zbl 1339.46028)],
as well as its VMO and CMO type subspaces. Some sufficient and necessary conditions for the inclusion and the continuity properties of \(T_g\) on these spaces are obtained.Pointwise characterizations of Besov and Triebel-Lizorkin spaces with generalized smoothness and their applicationshttps://zbmath.org/1496.460332022-11-17T18:59:28.764376Z"Li, Zi Wei"https://zbmath.org/authors/?q=ai:li.ziwei"Yang, Da Chun"https://zbmath.org/authors/?q=ai:yang.dachun.1|yang.dachun"Yuan, Wen"https://zbmath.org/authors/?q=ai:yuan.wen|yuan.wen.1The nowadays well-known homogeneous spaces \(\dot{A}^s_{p,q} (\mathbb R^n)\) with \(A \in \{B,F \}\), \(s\in \mathbb R\) and \(0<p,q \le \infty\) have been modified in several ways. The smoothness \(s\), characterized by \(\{ 2^{js} \}^\infty_{j=0}\), is generalized by suitable sequences \(\{\sigma_j \}^\infty_{j=0}\) of positive numbers. Furthermore, \(\mathbb R^n\) is replaced by metric spaces \((X, d, \mu)\) with the metric \(d\) and the Borel measure \(\mu\) on the set \(X\), where the smoothness is expressed by so-called Hajłasz gradients. More recently, there is some type of discretization, called hyperbolic filling. The paper deals with spaces based on these ingredients and their relations, especially to \(\dot{A}^\sigma_{p,q} (\mathbb R^n)\).
Reviewer: Hans Triebel (Jena)A note on embedding inequalities for weighted Sobolev and Besov spaceshttps://zbmath.org/1496.460342022-11-17T18:59:28.764376Z"Saito, Hiroki"https://zbmath.org/authors/?q=ai:saito.hirokiLet \(H^d\), \(0<d<n\), be the Hausdorff capacity in \(\mathbb R^n\). The limiting embeddings
\[
\int_{\mathbb R^n} |f| \, d H^{n-k} \le c \, \|f \, | \dot{W}^k_1 (\mathbb R^n) \|, \qquad f \in \mathscr D(\mathbb R^n),
\]
\(1\le k <n\), \(k\in \mathbb N\), for the related homogeneous Sobolev spaces and its generalization
\[
\int_{\mathbb R^n} |f| \, d H^{n-s} \le c \,\|f \, | \dot{B}^s_{1,1} (\mathbb R^n) \|, \qquad f\in \mathscr D(\mathbb R^n),
\]
\(0<s<n\), for the related homogeneous Besov spaces go back to \textit{D. R. Adams} [Lect. Notes Math. 1302, 115--124 (1988; Zbl 0658.31009)] and \textit{J. Xiao} [Adv. Math. 207, No. 2, 828--846 (2006; Zbl 1104.46022)]. The paper deals with weighted generalizations of these assertions both for weighted Hausdorff capacities and weighted Sobolev-Besov spaces where the weights belong to the Muckenhoupt class~\(A_1\).
Reviewer: Hans Triebel (Jena)Sobolev's inequality in central Herz-Morrey-Musielak-Orlicz spaces over metric measure spaceshttps://zbmath.org/1496.460352022-11-17T18:59:28.764376Z"Ohno, Takao"https://zbmath.org/authors/?q=ai:ohno.takao"Shimomura, Tetsu"https://zbmath.org/authors/?q=ai:shimomura.tetsuSummary: We give the boundedness of the Hardy-Littlewood maximal operator \(M_\lambda\), \(\lambda \geq 1\), on central Herz-Morrey-Musielak-Orlicz spaces \(\mathcal{H}^{\Phi,q,\omega}(X)\) over bounded non-doubling metric measure spaces and to establish a generalization of Sobolev's inequality for Riesz potentials \(I_{\alpha,\tau}\), \(f\), \(\tau \geq 1\), \(\alpha > 0\) of functions in such spaces. As an application and example, we obtain the boundedness of \(M_\lambda\) and \(I_{\alpha,\tau}\) for double phase functionals \(\Phi\) such that \(\Phi(x,t)=t^{p(x)}+a(x)t^{q(x)}\), \(x \in X\), \(t \geq 0\). These results are new even for the doubling metric measure setting.Distributions in \(\mathbb{R}^3\) with a thick curvehttps://zbmath.org/1496.460362022-11-17T18:59:28.764376Z"Yang, Yunyun"https://zbmath.org/authors/?q=ai:yang.yunyunSummary: A theory of distributions called ``thick distributions'' was developed to incorporate a point singularity in the test function space. In this present article we consider a more general situation where test functions are singular on a curve in \(\mathbb{R}^3\). We construct a topological vector space of such test functions and, by duality, the space \(\mathcal{D}_{\ast , C}^\prime( \mathbb{R}^3)\) of distributions that are thick on this curve. We study several operations, including partial differentiation. We introduce the notion of line thick delta functions which is a lifting of line delta functions to \(\mathcal{D}_{\ast , C}^\prime( \mathbb{R}^3)\). These new distributions, in particular thick line delta functions, may have applications in providing more accurate models to some problems from physics, biology and engineering where a ``line source'' or a ``tube source'' is present. As an example of such applications, we propose a more refined model of a growth factor's reaction and diffusion of a very thin blood capillary in a bulk tumor, and give a solution of the corresponding PDE.Localization for hyperbolic measures on infinite-dimensional spaceshttps://zbmath.org/1496.460372022-11-17T18:59:28.764376Z"Kalinin, A. N."https://zbmath.org/authors/?q=ai:kalinin.a-nSummary: Properties of the extreme points of families of concave measures on infinite-dimensional locally convex spaces are studied. The localization method is generalized to hyperbolic measures on Fréchet spaces.Isomorphisms of some algebras of analytic functions of bounded type on Banach spaceshttps://zbmath.org/1496.460382022-11-17T18:59:28.764376Z"Halushchak, S. I."https://zbmath.org/authors/?q=ai:halushchak.s-iSummary: The theory of analytic functions is an important section of nonlinear functional analysis. In many modern investigations topological algebras of analytic functions and spectra of such algebras are studied. In this work we investigate the properties of the topological algebras of entire functions, generated by countable sets of homogeneous polynomials on complex Banach spaces.
Let \(X\) and \(Y\) be complex Banach spaces. Let \(\mathbb{A}= \{A_1, A_2, \ldots, A_n, \ldots\}\) and \(\mathbb{P}=\{P_1, P_2, \ldots, P_n, \ldots \}\) be sequences of continuous algebraically independent homogeneous polynomials on spaces \(X\) and \(Y\), respectively, such that \(\|A_n\|_1=\|P_n\|_1=1\) and \(\deg A_n=\deg P_n=n\), \(n\in \mathbb{N}\). We consider the subalgebras \(H_{b\mathbb{A}}(X)\) and \(H_{b\mathbb{P}}(Y)\) of the Fréchet algebras \(H_b(X)\) and \(H_b(Y)\) of entire functions of bounded type, generated by the sets \(\mathbb{A}\) and \(\mathbb{P} \), respectively. It is easy to see that \(H_{b\mathbb{A}}(X)\) and \(H_{b\mathbb{P}}(Y)\) are the Fréchet algebras as well. In this paper we investigate conditions of isomorphism of the topological algebras \(H_{b\mathbb{A}}(X)\) and \(H_{b\mathbb{P}}(Y).\) We also present some applications for algebras of symmetric analytic functions of bounded type. In particular, we consider the subalgebra \(H_{bs}(L_{\infty})\) of entire functions of bounded type on \(L_{\infty}[0,1]\) which are symmetric, i.e. invariant with respect to measurable bijections of \([0,1]\) that preserve the measure. We prove that \(H_{bs}(L_{\infty})\) is isomorphic to the algebra of all entire functions of bounded type, generated by countable set of homogeneous polynomials on complex Banach space \(\ell_{\infty}\).A note on Huijsmans-de Pagter problem in ordered Banach algebrashttps://zbmath.org/1496.460392022-11-17T18:59:28.764376Z"Drnovšek, Roman"https://zbmath.org/authors/?q=ai:drnovsek.romanSummary: We give an example of a positive element \(a\) in some ordered Banach algebra \(\mathcal{A}\) such that its spectrum is equal to \(\{1\}\) and it is not greater than or equal to the unit element of \(\mathcal{A}\).Spectral operation in locally convex algebrashttps://zbmath.org/1496.460402022-11-17T18:59:28.764376Z"El Boukasmi, Driss"https://zbmath.org/authors/?q=ai:el-boukasmi.driss"El Kinani, Abdellah"https://zbmath.org/authors/?q=ai:el-kinani.abdellahLet \(A\) be an algebra and \(D\) a subset of the set \(\mathbb C\) of all complex numbers. A function \(f:D\to\mathbb C\) is said to operate on \(A\) if for each \(a\in A\), there exists a unique element \(a_f\in A\) such that the Fourier-Gelfand transform \(\widehat{a_f}\) of \(a_f\) is equal to the composition \(f\circ\hat{a}\) of the function \(f\) and the Fourier-Gelfand transform of \(a\). Moreover, a function \(f\) is said to operate spectrally on \(A\) if, for every \(a\in A\), such that the spectrum of \(a\) is included in \(D\), there exists a unique element \(b_f\in A\) such that the spectrum Sp\(_A(a_f)\) of \(a_f\) is equal to the image \(f(\mathrm{Sp}_A(a))\) of the spectrum of \(a\) by \(f\).
The authors find equivalent conditions for different kinds of algebras (including topological algebras) for the condition that all functions operate on \(A\) or operate spectrally on \(A\).
Reviewer: Mart Abel (Tartu)Topics on continuous inverse algebrashttps://zbmath.org/1496.460412022-11-17T18:59:28.764376Z"Naziri-Kordkandi, Ali"https://zbmath.org/authors/?q=ai:naziri-kordkandi.aliIn the setting of this paper, a continuous inverse algebra is a unital locally convex algebra in which the set of invertible elements is a neighbourhood of the unit element and inversion is continuous at the unit element. The results of the paper are mainly about the properties of topological zero divisors, the spectrum of an element, the spectral radius of an element in an algebra and about the multiplicativity of a linear map between continuous inverse algebras.
By Proposition 2.7 of the paper, one can see that every continuous inverse algebra is also a Waelbroeck algebra (i.e., an algebra, where the set of invertible elements is open and in which the inversion is continuous). But since Waelbroeck algebras do not have to be locally convex, the class of continuous inverse algebras is a proper subclass of Waelbroeck algebras. As continuous inverse algebras are also locally convex, they are also spectral algebras.
Some of the results are given for continuous inverse algebras, some for Banach algebras, some for Hausdorff algebras.
Nevertheless, several (if not all) results of this paper hold also in a more general setting, at least for all Waelbroeck algebras or even spectral algebras. For example, Theorem~3.9 holds for any algebra (see [\textit{T.~W. Palmer}, Banach algebras and the general theory of \(*\)-algebras. Volume~I: Algebras and Banach algebras. Paperback reprint of the 1994 hardback ed. Cambridge: Cambridge University Press (2009; Zbl 1176.46052)], p.~197, Proposition 2.1.8(a)); Theorem~3.5 holds for all Gelfand-Mazur algebras (definition of a Gelfand-Mazur algebra; Result 2.2.3 in [loc. cit.], pp.~211--212); Theorem~4.1 holds for all spectral algebras (Theorem 2.4.13 in [loc. cit.], p.~242, (f) implies~(d)).
Reviewer: Mart Abel (Tartu)Representations of (quasi-)complemented algebrashttps://zbmath.org/1496.460422022-11-17T18:59:28.764376Z"Haralampidou, Marina"https://zbmath.org/authors/?q=ai:haralampidou.marina"Tzironis, Konstantinos"https://zbmath.org/authors/?q=ai:tzironis.konstantinosThe authors continue their study of left complemented algebras.
In this paper, they offer some sufficient conditions when a Hausdorff left complemented or left quasi-complemented locally \(m\)-convex algebra has a continuous representation on a pre-Hilbert or a Hilbert space (Theorems \(6\), \(8\), \(18\), \(25\) and Corollaries \(26\), \(27\) and \(28\)).
Reviewer: Mart Abel (Tartu)Coproducts in the category Seg of Segal topological algebrashttps://zbmath.org/1496.460432022-11-17T18:59:28.764376Z"Abel, Mart"https://zbmath.org/authors/?q=ai:abel.martSummary: In this paper we find a sufficient condition for a family of Segal topological algebras to have a coproduct in the category Seg.Hilbert Arens algebra-modules and semigrouphttps://zbmath.org/1496.460442022-11-17T18:59:28.764376Z"Tao, Jicheng"https://zbmath.org/authors/?q=ai:tao.jicheng"Ai, Ying"https://zbmath.org/authors/?q=ai:ai.yingSummary: In this paper, we consider the semigroup of Hilbert Arens algebra-modules by using the semigroup theory and Hilbert Arens algebra-modules from [\textit{S. Cerreia-Vioglio} et al., J. Math. Anal. Appl. 446, No. 1, 970--1017 (2017; Zbl 1364.46044)]. We show when \(A\) is a finite dimensional Arens algebra and \(H\) is a Hilbert \(A\)-module, the semigroup \(\{(tT)\}_{t\geq 0}\subset L(H)\) is an \(m\)-continuous semigroup if and only if it is an \(H\)-continuous semigroup if and only if it is a \(\varphi\)-continuous semigroup.Almost multiplicative maps and \(\varepsilon\)-spectrum of an element in Fréchet \(Q\)-algebrahttps://zbmath.org/1496.460452022-11-17T18:59:28.764376Z"Farajzadeh, A. P."https://zbmath.org/authors/?q=ai:farajzadeh.ali-p"Omidi, M. R."https://zbmath.org/authors/?q=ai:omidi.mohammad-rezaSummary: Let \((A,(p_k))\) be a Fréchet \(Q\)-algebra with unit \(e_A\). The \(\varepsilon\)-spectrum of an element \(x\) in \(A\) is defined by
\[
\sigma_\varepsilon(x)=\left\{\lambda\in\mathbb{C}:p_{k_0}(\lambda e_A-x)p_{k_0}(\lambda e_A-x)^{-1}\geq \frac{1}{\varepsilon}\right\}
\]
for \(0<\varepsilon<1\). We show that there is a close relation between the
\(\varepsilon\)-spectrum and almost multiplicative maps. It is also shown that
\[
\{\varphi(x):\varphi\in M^\varepsilon_{alm}(A),\varphi(e_A)=1\}\subseteq\sigma_\varepsilon(x)
\]
for every \(x\in A\), where \(M^\varepsilon_{alm}(A)\) is the set of all \(\varepsilon\)-multiplicative maps from \(A\) to \(\mathbb{C}\).Certain properties of Jordan homomorphisms, \(n\)-Jordan homomorphisms and \(n\)-homomorphisms on rings and Banach algebrashttps://zbmath.org/1496.460462022-11-17T18:59:28.764376Z"Honary, Taher Ghasemi"https://zbmath.org/authors/?q=ai:ghasemi-honary.taherSummary: We investigate under what conditions \(n\)-Jordan homomorphisms between rings are \(n\)-homomorphism, or homomorphism; and under what conditions, \(n\)-Jordan homomorphisms are continuous.
One of the main goals in this work is to show that every \(n\)-Jordan homomorphism \(f : A \rightarrow B\), from a unital ring \(A\) into a ring \(B\) with characteristic greater than \(n\), is a multiple of a Jordan homomorphism and hence, it is an \(n\)-homomorphism if every Jordan homomorphism from \(A\) into \(B\) is a homomorphism. In particular, if \(B\) is an integral domain whose characteristic is greater than \(n\), then every \(n\)-Jordan homomorphism \(f : A \rightarrow B\) is an \(n\)-homomorphism.
Along with some other results, we show that if \(A\) and \(B\) are unital rings such that the characteristic of \(B\) is greater than \(n\), then every unital \(n\)-Jordan homomorphism \(f : A \rightarrow B\) is a Jordan homomorphism and hence, it is an \(m\)-Jordan homomorphism for any positive integer \(m \geq 2\).
We also investigate the automatic continuity of \(n\)-Jordan homomorphisms from a unital Banach algebra either into a semisimple commutative Banach algebra, onto a semisimple Banach algebra, or into a strongly semisimple Banach algebra whenever the \(n\)-Jordan homomorphism has dense range.Positivity and Schwarz inequality for Banach \(*\)-algebrashttps://zbmath.org/1496.460472022-11-17T18:59:28.764376Z"El Harti, Rachid"https://zbmath.org/authors/?q=ai:el-harti.rachid"Pinto, Paulo R."https://zbmath.org/authors/?q=ai:pinto.paulo-r-fSummary: We establish the Schwarz inequality for a class of Banach \(*\)-algebras, and use it to derive some consequences, for example when a linear map between Banach \(*\)-algebras is a Jordan homomorphism. This applies to the class of Banach \(*\)-algebras \(\ell^1 (G,A;\alpha)\) arising from \(C^*\)-dynamical systems \((A,G,\alpha)\) with \(\alpha\) an action of a discrete group \(G\) on a separable \(C^*\)-algebra \(A\).Problems of the algorithmization of algebraic systemshttps://zbmath.org/1496.460482022-11-17T18:59:28.764376Z"Ayupov, Sh. A."https://zbmath.org/authors/?q=ai:ayupov.shabkat-abdullaevich|ayupov.sh-a"Kabulov, V. K."https://zbmath.org/authors/?q=ai:kabulov.vasil-kabulovich(no abstract)The nuclear dimension of \(\mathcal{O}_\infty \)-stable \(C^\ast \)-algebrashttps://zbmath.org/1496.460492022-11-17T18:59:28.764376Z"Bosa, Joan"https://zbmath.org/authors/?q=ai:bosa.joan"Gabe, James"https://zbmath.org/authors/?q=ai:gabe.james"Sims, Aidan"https://zbmath.org/authors/?q=ai:sims.aidan"White, Stuart"https://zbmath.org/authors/?q=ai:white.stuart-aThe nuclear dimension for \(C^*\)-algebras provides a non-commutative analogue of the Lebesgue covering dimension. By tensorial absorption of \(\mathcal{O}_{\infty}\), it has been proved that the nuclear dimension of Kirchberg algebras is~$1$, which raises the natural question ``What is the nuclear dimension of \(\mathcal{O}_{\infty}\)-stable \(C^*\)-algebras?'' The authors give the answer by proving that separable, nuclear \(\mathcal{O}_{\infty}\)-stable \(C^*\)-algebras have nuclear dimension~$ 1$. They also determine when separable nuclear \(\mathcal{O}_{\infty}\)-stable \(C^*\)-algebras have finite decomposition rank, which precedes the notion of nuclear dimension.
Although originally defined for \(C^*\)-algebras, the definitions of both decomposition rank and nuclear dimension of a \(C^*\)-algebra \(A\) are given in terms of approximation properties for the identity map \(\mathrm{id}_A : A \to A\), and make perfect sense for other \(*\)-homomorphisms between \(C^*\)-algebras. The third main theorem proves that all \(\mathcal{O}_{\infty}\)-stable maps have nuclear dimension at most~$1$. Finally, the authors present a result to distinguish the nuclear dimensions of the full, \(\mathcal{O}_{2}\)-stable \(*\)-homomorphism between \(C^*\)-algebras \(A\) and \(B\) with \(A\) separable and exact. Especially, for the nuclear homomorphisms, they have nuclear dimension \(0\) (\(1\)) when the \(C^*\)-algebra \(A\) is (is not) quasidiagonal, and a nonnuclear homomorphism has infinite nuclear dimension.
Reviewer: Jingming Zhu (Jiaxing)Homotopy of product systems and \(K\)-theory of Cuntz-Nica-Pimsner algebrashttps://zbmath.org/1496.460502022-11-17T18:59:28.764376Z"Fletcher, James"https://zbmath.org/authors/?q=ai:fletcher.james"Gillaspy, Elizabeth"https://zbmath.org/authors/?q=ai:gillaspy.elizabeth"Sims, Aidan"https://zbmath.org/authors/?q=ai:sims.aidanThe authors study the \(K\)-theory of Cuntz-Nica-Pimsner algebras of product systems over \(\mathbb{N}^k\), with a particular focus on those built from higher-rank graphs (or \(k\)-graphs) and cubical \(2\)-cocycles. More precisely, the authors introduce a notion of a homotopy of product systems over a semigroup \(P\) with coefficient algebra \(A\), which is a nondegenerate product system over \(P\) with coefficient algebra \(\mathrm{C}([0,1],A)\) in which the canonical left and right actions of \(\mathrm{C}([0,1])\) on each fibre coincide. They prove that homotopy of product systems is an equivalence relation, that the Nica-Toeplitz algebras of homotopic product systems over \(\mathbb{N}^k\) are KK-equivalent under certain hypotheses, and that the Cuntz-Nica-Pimsner algebras of homotopic product systems over \(\mathbb{N}^k\) have isomorphic $K$-theory (and are KK-equivalent under certain conditions).
The authors then investigate unitary cocycles for \(k\)-graphs, which are in one-to-one correspondence with product systems over \(\mathbb{N}^k\) that can be constructed from \(k\)-graphs. For any row-finite, source-free \(k\)-graph \(\Lambda\) and cubical \(2\)-cocycle on \(\Lambda\), the authors show how to obtain a unitary cocycle such that the twisted \(C^*\)-algebra of the \(k\)-graph coincides with the Cuntz-Nica-Pimsner algebra of the product system determined by the unitary cocycle. The authors use this result to apply their main theorem on homotopies of product systems to twisted \(C^*\)-algebras of certain \(k\)-graphs. They prove that if there is a continuous path of unitaries between the unitary cocycles associated to two \(k\)-graphs with the same skeleton and \(2\)-cocycles on these graphs, then the associated product systems are homotopic, and hence the associated twisted \(k\)-graph \(C^*\)-algebras (or Cuntz-Nica-Pimsner algebras) have isomorphic \(K_0\) and \(K_1\) groups.
The authors also show that path-connectedness (that is, the existence of a continuous path of unitaries as described above) always holds in the case where \(k = 2\), and from this they deduce that the \(K\)-theory of a twisted \(2\)-graph \(C^*\)-algebra is independent of both the twisting \(2\)-cocycle and the factorisation rules of the graph. In particular, they provide a new proof of Evans' theorem (see [\textit{D. G. Evans}, New York J. Math. 14, 1--31 (2008; Zbl 1146.46048)]) that the \(K\)-theory of the \(C^*\)-algebra of a row-finite, source-free \(2\)-graph depends only on the skeleton of the graph (and not on the factorisation rules).
Finally, the authors investigate the condition of path-connectedness of the unitary cocycles for \(k\)-graphs with a single vertex, and they reduce the question of whether the \(K\)-theory of a twisted \(k\)-graph \(C^*\)-algebra is independent of the twisting \(2\)-cocycle and of the factorisation rules of the graph to a question of path-connectedness of the space of solutions to an equation of Yang-Baxter type.
Reviewer: Becky Armstrong (Münster)The Haagerup property for twisted groupoid dynamical systemshttps://zbmath.org/1496.460512022-11-17T18:59:28.764376Z"Kwaśniewski, Bartosz K."https://zbmath.org/authors/?q=ai:kwasniewski.bartosz-kosma"Li, Kang"https://zbmath.org/authors/?q=ai:li.kang"Skalski, Adam"https://zbmath.org/authors/?q=ai:skalski.adam-gSummary: We introduce the Haagerup property for twisted groupoid \(C^\ast\)-dynamical systems in terms of naturally defined positive-definite operator-valued multipliers. By developing a version of `the Haagerup trick' we prove that this property is equivalent to the Haagerup property of the reduced crossed product \(C^\ast\)-algebra with respect to the canonical conditional expectation \(E\). This extends a theorem of Dong and Ruan [\textit{Z.~Dong} and \textit{Z.-J. Ruan}, Integral Equations Oper. Theory 73, No.~3, 431--454 (2012; Zbl 1263.46043)]
for discrete group actions, and implies that a given Cartan inclusion of separable \(C^\ast\)-algebras has the Haagerup property if and only if the associated Weyl groupoid has the Haagerup property in the sense of Tu [\textit{J.-L. Tu}, \(K\)-Theory 17, No. 3, 215--264 (1999; Zbl 0939.19001)]. We use the latter statement to prove that every separable \(C^\ast\)-algebra which has the Haagerup property with respect to some Cartan subalgebra satisfies the Universal Coefficient Theorem. This generalises a recent result of Barlak and Li [\textit{S.~Barlak} and \textit{X.~Li}, Adv. Math. 316, 748--769 (2017; Zbl 1382.46048)] on the UCT for nuclear Cartan pairs.Self-similar \(k\)-graph \(C^*\)-algebrashttps://zbmath.org/1496.460522022-11-17T18:59:28.764376Z"Li, Hui"https://zbmath.org/authors/?q=ai:li.hui"Yang, Dilian"https://zbmath.org/authors/?q=ai:yang.dilianSummary: In this paper, we introduce a notion of a self-similar action of a group \(G\) on a \(k\)-graph \(\Lambda\) and associate it a universal \(C^\ast\)-algebra \(\mathcal{O}_{G,\Lambda}\). We prove that \(\mathcal{O}_{G,\Lambda}\) can be realized as the Cuntz-Pimsner algebra of a product system. If \(G\) is amenable and the action is pseudo free, then \(\mathcal{O}_{G,\Lambda}\) is shown to be isomorphic to a ``path-like'' groupoid \(C^\ast\)-algebra. This facilitates studying the properties of \(\mathcal{O}_{G,\Lambda}\). We show that \(\mathcal{O}_{G,\Lambda}\) is always nuclear and satisfies the universal coefficient theorem; we characterize the simplicity of \(\mathcal{O}_{G,\Lambda}\) in terms of the underlying action, and we prove that, whenever \(\mathcal{O}_{G,\Lambda}\) is simple, there is a dichotomy: it is either stably finite or purely infinite, depending on whether \(\Lambda\) has nonzero graph traces or not. Our main results generalize the recent work of Exel and Pardo [\textit{R.~Exel} and \textit{E.~Pardo}, Adv. Math. 306, 1046--1129 (2017; Zbl 1390.46050)]
on self-similar graphs.Strongly peaking representations and compressions of operator systemshttps://zbmath.org/1496.460532022-11-17T18:59:28.764376Z"Davidson, Kenneth R."https://zbmath.org/authors/?q=ai:davidson.kenneth-r"Passer, Benjamin"https://zbmath.org/authors/?q=ai:passer.benjamin-wSummary: We use Arveson's notion of strongly peaking representation to generalize uniqueness theorems for free spectrahedra and matrix convex sets that admit minimal presentations. A fully compressed separable operator system necessarily generates the \(C^*\)-envelope and is such that the identity is the direct sum of strongly peaking representations. In particular, a fully compressed presentation of a separable operator system is unique up to unitary equivalence. Under various additional assumptions, minimality conditions are sufficient to determine a separable operator system uniquely.Noncommutative Choquet simpliceshttps://zbmath.org/1496.460542022-11-17T18:59:28.764376Z"Kennedy, Matthew"https://zbmath.org/authors/?q=ai:kennedy.matthew"Shamovich, Eli"https://zbmath.org/authors/?q=ai:shamovich.eliSummary: We introduce a notion of noncommutative Choquet simplex, or briefly an nc simplex, that generalizes the classical notion of a simplex. While every simplex is an nc simplex, there are many more nc simplices. They arise naturally from \(C^*\)-algebras and in noncommutative dynamics. We characterize nc simplices in terms of their geometry and in terms of structural properties of their corresponding operator systems. There is a natural definition of nc Bauer simplex that generalizes the classical definition of a Bauer simplex. We show that a compact nc convex set is an nc Bauer simplex if and only if it is affinely homeomorphic to the nc state space of a unital \(C^*\)-algebra, generalizing a classical result of Bauer for unital commutative \(C^*\)-algebras. We obtain several applications to noncommutative dynamics. We show that the set of nc states of a \(C^*\)-algebra that are invariant with respect to the action of a discrete group is an nc simplex. From this, we obtain a noncommutative ergodic decomposition theorem with uniqueness. Finally, we establish a new characterization of discrete groups with Kazhdan's property (T) that extends a result of \textit{E. Glasner} and \textit{B. Weiss} [Geom. Funct. Anal. 7, No. 5, 917--935 (1997; Zbl 0899.22006)]. Specifically, we show that a discrete group has property (T) if and only if for every action of the group on a unital \(C^*\)-algebra, the set of invariant states is affinely homeomorphic to the state space of a unital \(C^*\)-algebra.Corrigendum to: ``Dual spaces of operator systems''https://zbmath.org/1496.460552022-11-17T18:59:28.764376Z"Ng, Chi-Keung"https://zbmath.org/authors/?q=ai:ng.chi-keungFrom the text: All the operator systems considered in the author's paper [ibid. 508, No. 2, Article ID 125890, 23 p. (2022; Zbl 1492.46051)] need be assumed to be complete. The precise changes can be found in the most updated arXiv version [\textit{C. K. Ng}, ``Duality of operator systems'''', Preprint (2022), \url{arXiv:2105.11112v3}].Unitary conjugacy for type III subfactors and \(W^\ast\)-superrigidityhttps://zbmath.org/1496.460562022-11-17T18:59:28.764376Z"Isono, Yusuke"https://zbmath.org/authors/?q=ai:isono.yusukeThe present article establishes a new criterion for intertwining by bimodules à la Popa in the setting of type III factors. Its main result in this direction, which is contained in Theorem A and Corollary B, states that for arbitrary inclusions of \(\sigma\)-finite von Neumann algebras \(A, B \subset M\) with conditional expectations \(\mathrm{E}_A \colon 1_A M 1_A \to A\) and \(\mathrm{E}_B \colon 1_B M 1_B \to B\), respectively, intertwining \(A \preceq_M B\) is equivalent to intertwining of continuous cores in the following sense. If \((N, \omega)\) is any fixed type III\(_1\) factor with a faithful normal state, both independent of \(A\), \(B\) and \(M\), and \(\varphi, \psi \in M_*\) are faithful normal states preserved by \(\mathrm{E}_B\) and \(\mathrm{E}_A\), respectively, then intertwining of \(A\) into \(B\) inside \(M\) is equivalent to the condition
\begin{gather*}
\Pi(\mathrm{C}_{\psi \otimes \omega}(A \overline{\otimes} N)) \preceq_{\mathrm{C}_{\varphi \otimes \omega}(M \overline{\otimes} N)} \mathrm{C}_{\varphi \otimes \omega}(B \overline{\otimes} N) \text{.}
\end{gather*}
This result allows the author to employ known characterisations of intertwining in the setup of type II von Neumann algebras. In this way, he achieves a \(W^\ast\)-superrigidity result for crossed products associated with Bernoulli shifts whose base is an amenable III\(_1\) factor among all crossed product von Neumann algebras arising from state-preserving, outer actions of discrete groups on amenable factors. Further, a stable strong solidity result for arbitrary free product von Neumann algebras is obtained.
Reviewer: Sven Raum (Stockholm)A generalized powers averaging property for commutative crossed productshttps://zbmath.org/1496.460572022-11-17T18:59:28.764376Z"Amrutam, Tattwamasi"https://zbmath.org/authors/?q=ai:amrutam.tattwamasi"Ursu, Dan"https://zbmath.org/authors/?q=ai:ursu.danPowers' averaging property for discrete groups has played an important role in questions about simplicity related to reduced group \(C^\ast\)-algebras and reduced crossed products. In the present paper, the authors introduce a generalized version of Powers' averaging property for reduced crossed products of group \(C^\ast\)-algebras, and prove that it is equivalent to simplicity of the crossed product.
Reviewer: Luoyi Shi (Tianjin)Stable elements and property (S)https://zbmath.org/1496.460582022-11-17T18:59:28.764376Z"Bosa, Joan"https://zbmath.org/authors/?q=ai:bosa.joanSummary: We study the relation (and differences) between stability and Property (S) in the simple and stably finite framework. This leads us to characterize stable elements in terms of their support, and study these concepts from different sides: hereditary subalgebras, projections in the multiplier algebra and order properties in the Cuntz semigroup. We use these approaches to show both that cancellation at infinity on the Cuntz semigroup just holds when its Cuntz equivalence is given by isomorphism at the level of Hilbert right-modules, and that different notions such as Regularity, \(\omega\)-comparison, Corona Factorization Property, property R, etc. are equivalent under mild assumptions.Unitary Cuntz semigroups of ideals and quotientshttps://zbmath.org/1496.460592022-11-17T18:59:28.764376Z"Cantier, Laurent"https://zbmath.org/authors/?q=ai:cantier.laurentThis paper is about unitary Cuntz semigroups, a modified version of Cuntz semigroups introduced by the author in order to capture \(K_1\) information. The author explains how the unitary Cuntz semigroup encodes the ideal structure of \(C^*\)-algebras. Moreover, the author also establishes a precise connection between the unitary Cuntz semigroup on the one hand and the original Cuntz semigroup and \(K_1\) on the other hand.
Reviewer: Xin Li (Glasgow)Approximately multiplicative decompositions of nuclear mapshttps://zbmath.org/1496.460602022-11-17T18:59:28.764376Z"Wagner, Douglas"https://zbmath.org/authors/?q=ai:wagner.douglas-aSummary: We expand upon work from many hands on the decomposition of nuclear maps. Such maps can be characterised by their ability to be approximately written as the composition of maps to and from matrices. Under certain conditions (such as quasidiagonality), we can find a decomposition whose maps behave nicely, by preserving multiplication up to an arbitrary degree of accuracy and being constructed from order-zero maps (as in the definition of nuclear dimension). We investigate these conditions and relate them to a \(\mathrm{W}^*\)-analogue.Modular structure of the Weyl algebrahttps://zbmath.org/1496.460612022-11-17T18:59:28.764376Z"Longo, Roberto"https://zbmath.org/authors/?q=ai:longo.robertoSummary: We study the modular Hamiltonian associated with a Gaussian state on the Weyl algebra. We obtain necessary/sufficient criteria for the local equivalence of Gaussian states, independently of the classical results by Araki and Yamagami, Van Daele, Holevo. We also present a criterion for a Bogoliubov automorphism to be weakly inner in the GNS representation. The main application of our analysis is the description of the vacuum modular Hamiltonian associated with a time-zero interval in the scalar, massive, free QFT in two spacetime dimensions, thus complementing the recent results in higher space dimensions [\textit{R.~Longo} and \textit{G.~Morsella}, ``The massless modular Hamiltonian'', Preprint (2020), \url{arXiv:2012.00565}]. In particular, we have the formula for the local entropy of a one-dimensional Klein-Gordon wave packet and Araki's vacuum relative entropy of a coherent state on a double cone von Neumann algebra. Besides, we derive the type {III}\(_1\) factor property. Incidentally, we run across certain positive selfadjoint extensions of the Laplacian, with outer boundary conditions, seemingly not considered so far.Essentially invertible measurable operators affiliated to a semifinite von Neumann algebra and commutatorshttps://zbmath.org/1496.460622022-11-17T18:59:28.764376Z"Bikchentaev, A. M."https://zbmath.org/authors/?q=ai:bikchentaev.airat-mSummary: Suppose that a von Neumann operator algebra \({\mathcal{M}}\) acts on a Hilbert space \({\mathcal{H}}\) and \(\tau\) is a faithful normal semifinite trace on \({\mathcal{M}} \). If Hermitian operators \(X,Y\in S({\mathcal{M}},\tau)\) are such that \(-X\leq Y\leq X\) and \(Y\) is \(\tau \)-essentially invertible then so is \(X \). Let \(0<p\leq 1 \). If a \(p \)-hyponormal operator \(A\in S({\mathcal{M}},\tau)\) is right \(\tau \)-essentially invertible then \(A\) is \(\tau \)-essentially invertible. If a \(p \)-hyponormal operator \(A\in{\mathcal{B}}({\mathcal{H}})\) is right invertible then \(A\) is invertible in \({\mathcal{B}}({\mathcal{H}}) \). If a hyponormal operator \(A\in S({\mathcal{M}},\tau)\) has a right inverse in \(S({\mathcal{M}},\tau)\) then \(A\) is invertible in \(S({\mathcal{M}},\tau) \). If \(A,T\in{\mathcal{M}}\) and \(\mu_t(A^n)^{\frac{1}{n}}\to 0\) as \(n\to\infty\) for every \(t>0\) then \(AT ( TA )\) has no right (left) \( \tau \)-essential inverse in \(S({\mathcal{M}},\tau) \). Suppose that \({\mathcal{H}}\) is separable and \(\dim{\mathcal{H}}=\infty \). A right (left) essentially invertible operator \(A\in{\mathcal{B}}({\mathcal{H}})\) is a commutator if and only if the right (left) essential inverse of \(A\) is a commutator.On entropy for general quantum systemshttps://zbmath.org/1496.460632022-11-17T18:59:28.764376Z"Majewski, W. A."https://zbmath.org/authors/?q=ai:majewski.wladyslaw-adam"Labuschagne, L. E."https://zbmath.org/authors/?q=ai:labuschagne.louis-eSummary: In these notes we will give an overview and road map for a definition and characterization of (relative) entropy for both classical and quantum systems. In other words, we will provide a consistent treatment of entropy which can be applied within the recently developed Orlicz space based approach to large systems. This means that the proposed approach successfully provides a refined framework for the treatment of entropy in each of classical statistical physics, Dirac's formalism of Quantum Mechanics, large systems of quantum statistical physics, and finally also for Quantum Field Theory.Extreme point methods in the study of isometries on certain noncommutative spaceshttps://zbmath.org/1496.460642022-11-17T18:59:28.764376Z"De Jager, Pierre"https://zbmath.org/authors/?q=ai:de-jager.pierre"Conradie, Jurie"https://zbmath.org/authors/?q=ai:conradie.jurieSummary: In this paper, we characterize surjective isometries on certain classes of noncommutative spaces associated with semi-finite von Neumann algebras: the Lorentz spaces \(L^{w,1}\), as well as the spaces \(L^1+L^\infty\) and \(L^1\cap L^\infty \). The technique used in all three cases relies on characterizations of the extreme points of the unit balls of these spaces. Of particular interest is that the representations of isometries obtained in this paper are global representations.Limit laws for $R$-diagonal variables in a tracial probability spacehttps://zbmath.org/1496.460652022-11-17T18:59:28.764376Z"Zhou, Cong"https://zbmath.org/authors/?q=ai:zhou.congSummary: We study the weak convergence of sums of \(\ast\)-free, identically distributed tracial \(R\)-diagonal variables. The result parallels earlier results about free additive convolution on the real line. In particular, we determine under which conditions an infinitesimal array yields a sequence that converges to a given infinitely divisible tracial \(R\)-diagonal distribution. Thus, much of the work concerning sums of free (in the sense of Voiculescu) identically distributed positive random variables can be translated to the tracial \(R\)-diagonal context.Series of free \(R\)-diagonal random variableshttps://zbmath.org/1496.460662022-11-17T18:59:28.764376Z"Bercovici, Hari"https://zbmath.org/authors/?q=ai:bercovici.hari"Nica, Alexandru"https://zbmath.org/authors/?q=ai:nica.alexandru"Zhong, Ping"https://zbmath.org/authors/?q=ai:zhong.pingSummary: We prove an analogue of Kolmogorov's three series theorem for a series of free \(R\)-diagonal random variables. Namely, a series of free \(R\)-diagonal random variables in a tracial probability space converges almost uniformly if and only if two numerical series converge. We also introduce a convolution operation \(\boxplus_{RD}\) for probability measures on \([0,+\infty)\). This reflects the operation of addition for free \(R\)-diagonal elements in a noncommutative probability space. It turns out that an iteration of the \(R\)-transform linearizes the \(\boxplus_{RD}\) convolution and is instrumental in our arguments.Conditional expectation, entropy, and transport for convex Gibbs laws in free probabilityhttps://zbmath.org/1496.460672022-11-17T18:59:28.764376Z"Jekel, David"https://zbmath.org/authors/?q=ai:jekel.davidSummary: Let \((X_1,\dots ,X_m)\) be self-adjoint noncommutative random variables distributed according to the free Gibbs law given by a sufficiently regular convex and semi-concave potential \(V\), and let \((S_1,\dots,S_m)\) be a free semicircular family. For \(k<m\), we show that conditional expectations and conditional non-microstates free entropy given \(X_1,\dots,X_k\) arise as the large \(N\) limit of the corresponding conditional expectations and entropy for the \(N\times N\) random matrix models associated to \(V\). Then, by studying conditional transport of measure for the matrix models, we construct an isomorphism \(\mathrm{W}^*(X_1,\dots,X_m)\to\mathrm{W}^*(S_1,\dots,S_m)\) that maps \(\mathrm{W}^*(X_1,\dots,X_k)\) to \(\mathrm{W}^*(S_1,\dots,S_k)\) for each \(k=1,\dots,m\) and that also witnesses the Talagrand inequality for the law of \((X_1,\dots,X_m)\) relative to the law of \((S_1,\dots,S_m)\).Strongly outer actions of amenable groups on \(\mathcal{Z}\)-stable nuclear \(C^\ast\)-algebrashttps://zbmath.org/1496.460682022-11-17T18:59:28.764376Z"Gardella, Eusebio"https://zbmath.org/authors/?q=ai:gardella.eusebio"Hirshberg, Ilan"https://zbmath.org/authors/?q=ai:hirshberg.ilan"Vaccaro, Andrea"https://zbmath.org/authors/?q=ai:vaccaro.andreaThe paper under review concerns the action \(\alpha\) of a discrete, countable, amenable group \(G\) on a separable, simple, unital, nuclear, \(\mathcal{Z}\)-stable \(C^*\)-algebra \(A\). It is motivated by researching general conditions for the preservation of \(\mathcal{Z}\)-stability by the induced crossed product \(A \rtimes G\). Towards this end, the authors study the relation of strong outerness of \(\alpha\) to two weaker versions of the tracial Rohklin property for the stabilization \(\alpha \otimes\mathrm{id}_{\mathcal{Z}}\). The first one is the weak tracial Rohklin property that replaces projections with positive elements and has been under considerable study in the literature. The second concerns the Rohklin dimenion introduced by \textit{I. Hirshberg} et al. [Commun. Math. Phys. 335, No. 2, 637--670 (2015; Zbl 1333.46055)]. Furthermore, they provide conditions for \(\alpha\) to be cocycle conjugate to \(\alpha \otimes\mathrm{id}_{\mathcal{Z}}\), even when \(\alpha\) is not strongly outer. The paper under review contains two main results in this endeavour (under the hypotheses oulined on \(G\) and \(A\)).
First, if the orbits of the action induced by \(\alpha\) on \(T(A)\) are finite and their cardinality is uniformly bounded, they show that the strong outerness of \(\alpha\) is equivalent to the weak tracial Rohklin property for \(\alpha \otimes\mathrm{id}_{\mathcal{Z}}\); if, in particular, \(G\) is residually finite then the above are also equivalent to \(\alpha \otimes \mathrm{id}_{\mathcal{Z}}\) having finite Rohklin dimension (in fact at most \(2\)). The first equivalence extends previous results of \textit{S. Echterhoff} et al. [J. Reine Angew. Math. 639, 173--221 (2010; Zbl 1202.46081)], \textit{H. Matui} and \textit{Y. Sato} [Am. J. Math. 136, No. 6, 1441--1496 (2014; Zbl 1317.46042)], and \textit{Q.-Y. Wang} [Rocky Mt. J. Math. 48, No. 4, 1307--1344 (2018; Zbl 1408.46066)], by removing all smallness assumptions on the size of the orbits by the induced action on \(T(A)\) (modulo the uniform boundedness assumption). The second equivalence generalizes results of Liao for \(\mathbb{Z}^m\)-actions, see [\textit{H.-C. Liao}, J. Funct. Anal. 270, No. 10, 3675--3708 (2016; Zbl 1355.46055); Int. J. Math. 28, No. 7, Article ID 1750050, 22 p. (2017; Zbl 1383.46053)].
Secondly, if \(\partial_e T(A)\) is compact with \(\dim(\partial_e T(A)) < \infty\), the orbits of the induced action of \(\alpha\) on \(\partial_e T(A)\) are finite with uniformly bounded cardinality, and the orbit space \(\partial_e T(A)/ G\) is Hausdorff, then \(\alpha\) is cocycle conjugate to \(\alpha \otimes\mathrm{id}_{\mathcal{Z}}\). The conditions cover for example the case when the action induced on \(\partial_e T(A)\) factors through a finite group action. This generalizes results of \textit{Y. Sato} [Adv. Stud. Pure Math. 80, 189--210 (2019; Zbl 1434.46039)], and in turn of Matui and Sato [loc.~cit..], by weakening the assumption of a trivial action on the trace space.
In the process, the authors develop equivariant versions of complemented partitions of unity and uniform property \(\Gamma\), which are of independent interest. They show that if \(A\) is a separable, unital, nuclear \(C^*\)-algebra with non-empty trace space and with no finite-dimensional quotients, and the induced action of \(\alpha\) on \(T(A)\) has finite orbits uniformly bounded in size by \(M >0\), then: \((A, \alpha)\) has uniform property \(\Gamma\) if and only if \((A, \alpha)\) has complemented partitions of unity with constant \(M\), if and only if for every \(n \in \mathbb{N}\), there is a unital embedding of the matrix algebra \(M_n \to (A^{\mathcal{U}} \cap A') ^{\alpha^{\mathcal{U}}}\). If \(A\) is also \(\mathcal{Z}\)-stable and simple, then the above are equivalent to \((A, \alpha)\) being cocycle conjugate to \((A \otimes \mathcal{Z}, \alpha \otimes \mathrm{id}_{\mathcal{Z}})\).
Reviewer: Evgenios Kakariadis (Newcastle upon Tyne)A \(\mathbb{Z}_2\)-topological index for quasi-free fermionshttps://zbmath.org/1496.460692022-11-17T18:59:28.764376Z"Aza, N. J. B."https://zbmath.org/authors/?q=ai:aza.n-j-b"Reyes-Lega, A. F."https://zbmath.org/authors/?q=ai:reyes-lega.andres-f"Sequera, L. A. M."https://zbmath.org/authors/?q=ai:sequera.l-a-mSummary: We use infinite dimensional self-dual CAR \(C^*\)-algebras to study a \(\mathbb{Z}_2\)-index, which classifies free-fermion systems embedded on \(\mathbb{Z}^d\) disordered lattices. Combes-Thomas estimates are pivotal to show that the \(\mathbb{Z}_2\)-index is uniform with respect to the size of the system. We additionally deal with the set of ground states to completely describe the mathematical structure of the underlying system. Furthermore, the \(\mathrm{weak}^*\)-topology of the set of linear functionals is used to analyze paths connecting different sets of ground states.Induced coactions along a homomorphism of locally compact quantum groupshttps://zbmath.org/1496.460702022-11-17T18:59:28.764376Z"Kitamura, Kan"https://zbmath.org/authors/?q=ai:kitamura.kanSummary: We consider induced coactions on \(C^*\)-algebras along a homomorphism of locally compact quantum groups which need not give a closed quantum subgroup. Our approach generalizes the induced coactions constructed by Vaes, and also includes certain fixed point algebras. We focus on the case when the homomorphism satisfies a quantum analogue of properness. Induced coactions along such a homomorphism still admit the natural formulations of various properties known in the case of a closed quantum subgroup, such as imprimitivity and adjointness with restriction. Also, we show a relationship of induced coactions and restriction which is analogous to base change formula for modules over algebras. As an application, we give an example that shows several kinds of 1-categories of coactions with forgetful functors cannot recover the original quantum group.The Bergmann-Shilov boundary of a bounded symmetric domainhttps://zbmath.org/1496.460712022-11-17T18:59:28.764376Z"Mackey, M."https://zbmath.org/authors/?q=ai:mackey.michael-t"Mellon, P."https://zbmath.org/authors/?q=ai:mellon.paulineThe authors study the infinite-dimensional analogs of determining sets and Bergman-Shilov boundary for holomorphic maps in the setting of bounded symmetric domains in complex Banach spaces. Their approach relies upon \(\mathrm{JB}^*\)-triple algebraic techniques, based on the fact that any such domain is biholomorphically equivalent to the unit ball \(B\) of some \(\mathrm{JB}^*\)-triple \(E\) and any biholomorphic automorphism \(g\in\mathrm{Aut}(B)\) of \(B\) extends continuously to a neighborhood of the closure \(\overline{B}\).
After a brief introduction to \(\mathrm{JB}^*\)-theory providing a geometrically useful selection of Jordan-triple identities, they prove that the family \(\Gamma\) of the extreme points of \(\overline{B}\) resp. the family \(\Gamma_1\) of unitary tripotents is invariant under the maps \(g\in\mathrm{Aut}(B)\). Actually, the interesting results are achieved for the cases when \(\Gamma\ne\emptyset\) or \(\Gamma_1\ne\emptyset\). In particular, under such hypothesis we have a Russo-Dye type theorem, that is, \(\overline{B}\) is the closed convex hull of \(\Gamma\) resp. \(\Gamma_1\). Moreover, given any map \(f \in\mathrm{Hol}(\overline{B},X)\) into a complex Banach space \(X\) (\(f:\overline{B}\to X\) is continuous and holomorphic when restricted to \(B\)), we have \(f(\overline{B}) \subset \overline{\mathrm{co}}\big( f(\Gamma)\big)\) with \(\Vert f\Vert_{\overline{B}} = \Vert f\Vert_{\Gamma}\) resp. \(\Vert f\Vert_{\overline{B}} = \Vert f\Vert_{\Gamma_1}\) (for the sup-norms \(\Vert f \Vert_\Delta =\sup_{x\in\Delta} \Vert f(x) \Vert\)).
The major part of the paper is focused on the case of finte rank \(\mathrm{JB}^*\)-triples. The approach does not use their classification stating that we can regard \(E\) as a finite direct sum of reflexive Cartan factors, instead a unifying algebraic treatment is developed along the lines of some ideas in [\textit{W. Kaup} and \textit{J. Sauter}, Manuscr. Math. 101, No. 3, 351--360 (2000; Zbl 0981.32012)] on the structure of boundary components. The main result concerns the Bergman-Shilov boundary: If the underlying \(\mathrm{JB}^*\)-triple is of finite rank, then \(\Gamma\) is the smallest closed subset \(\Lambda\) of \(\overline{B}\) such that \(\Vert f\Vert_{\overline{B}} = \Vert f\Vert_{\Lambda}\) for all \(f\in\mathrm{Hol}(\overline{B},\mathbb{C})\).
Reviewer: László Stachó (Szeged)Baum-Connes and the Fourier-Mukai transformhttps://zbmath.org/1496.460722022-11-17T18:59:28.764376Z"Emerson, Heath"https://zbmath.org/authors/?q=ai:emerson.heath"Hudson, Daniel"https://zbmath.org/authors/?q=ai:hudson.danielIf \(\mathbb{T}^d = \mathbb{R}^d/\mathbb{Z}^d\) is the torus, the Poincaré bundle \(\mathcal{P}_d\) is a complex line bundle over \(\mathbb{T}^d \times \widehat{\mathbb{Z}}^d\) (where \(\widehat{\mathbb{Z}}^d = \Hom(\mathbb{Z}^d,\mathbb{T})\)). The Fourier-Mukai correspondence is the topological correspondence
\[
\mathbb{T}^d \xleftarrow{\text{pr}_1} (\mathbb{T}^d\times \widehat{\mathbb{Z}}^d, \mathcal{P}_d) \xrightarrow{\text{pr}_2} \widehat{\mathbb{Z}}^d,
\]
which defines an element \([\mathcal{F}_d] \in KK_{-d}(\mathbb{T}^d, \widehat{\mathbb{Z}}^d)\). The Fourier-Mukai transform is the map \([\mathcal{F}_d]\otimes_{\widehat{\mathbb{Z}}^d} : K^{\ast}(\widehat{\mathbb{Z}}^d) \to\) \(K^{\ast -d}(\mathbb{T}^d)\), and the first main result of the paper is a geometric description of this map.
Given a torsion-free discrete group \(G\) with classifying space \(BG\), the Baum-Connes assembly map
\[
\mu : K_{\ast}(BG) \to K_{\ast}(C^{\ast}(G))
\]
is given by
\[
\mu(f) = \mathcal{P}_G\otimes_{C(G)\otimes C^{\ast}(G)}(f\otimes 1_{C^{\ast}(G)})
\]
where \(\mathcal{P}_G \in KK_0(\mathbb{C}, C(BG)\otimes C^{\ast}(G))\) is the class of the Mischenko element (see [\textit{P. Baum} et al., Contemp. Math. 167, 241--291 (1994; Zbl 0830.46061)]). If \(G = \mathbb{Z}^d\), then \(\mathcal{P}_G\) is the class of the Poincaré bundle and the second main result of the paper is a geometric description of the assembly map in this case.
Reviewer: Prahlad Vaidyanathan (Bhopal)Dimension groups for self-similar maps and matrix representations of the cores of the associated \(C^*\)-algebrashttps://zbmath.org/1496.460732022-11-17T18:59:28.764376Z"Kajiwara, Tsuyoshi"https://zbmath.org/authors/?q=ai:kajiwara.tsuyoshi"Watatani, Yasuo"https://zbmath.org/authors/?q=ai:watatani.yasuoGiven a self-similar map, the authors construct a \(C^*\)-correspondence and study its Cuntz-Pimsner \(C^*\)-algebra. This \(C^*\)-algebra comes with a canonical gauge action, and the core is given by the corresponding fixed point algebra. The authors analyse this fixed point algebra in detail and discover an interesting connection to branched points. They also study \(K\)-theoretic invariants consisting of dimension groups and canonical endomorphisms. Several concrete examples are discussed.
Reviewer: Xin Li (Glasgow)The category of compact quantum metric spaceshttps://zbmath.org/1496.460742022-11-17T18:59:28.764376Z"Long, Botao"https://zbmath.org/authors/?q=ai:long.botao"Wu, Wei"https://zbmath.org/authors/?q=ai:wu.wei.3Characterization of homological properties of \(\theta\)-Lau product of Banach algebrashttps://zbmath.org/1496.460752022-11-17T18:59:28.764376Z"Essmaili, Morteza"https://zbmath.org/authors/?q=ai:essmaili.morteza"Rejali, Ali"https://zbmath.org/authors/?q=ai:rejali.ali"Marzijarani, Azam Salehi"https://zbmath.org/authors/?q=ai:marzijarani.azam-salehiSummary: Let \(A\) and \(B\) be two Banach algebras and \(\theta\in\sigma(B)\). In this paper, we investigate biprojectivity and biflatness of \(\theta\)-Lau product of Banach algebras \(A\times_\theta B\). Indeed, we show that \(A\times_\theta B\) is biprojective if and only if \(A\) is contractible and \(B\) is biprojective. This generalizes some known results. Moreover, we characterize biflatness of \(\theta\)-Lau product of Banach algebras under some conditions. As an application, we give an example of biflat Banach algebras \(A\) and \(X\) such that the generalized module extension Banach algebra \(X\rtimes A\) is not biflat. Finally, we characterize pseudo-contractibility of \(\theta\)-Lau product of Banach algebras and give an affirmative answer to an open question.\(\mathcal{L}\)-semireflexive subcategorieshttps://zbmath.org/1496.460762022-11-17T18:59:28.764376Z"Botnaru, Dumitru"https://zbmath.org/authors/?q=ai:botnaru.dumitruRésumée: Si \((\mathcal{K,L})\) est un couple de sous-catégories conjuguées de la catégorie des espaces localement convexes topologiques vectoriels Hausdorff, alors les catégories \(\mathcal{K}\) et \(\mathcal{L}\) sont isomorphes. Ainsi:
\begin{itemize}
\item[1.] Les lattices \(\mathbb{R}(\mathcal{K})\) et \(\mathbb{R}(\mathcal{L})\) des sous-catégories réflectives des catégories \(\mathcal{K}\)
et \(\mathcal{L}\) sont isomorphes.
\item[2.] Les lattices \(\mathbb{K}(\mathcal{K})\) et \(\mathbb{K}(\mathcal{L})\) des sous-catégories coréflectives des catégories
\(\mathcal{K}\) et \(\mathcal{L}\) sont isomorphes.
\end{itemize}
En construisant l'isomorphisme des lattices \(\mathbb{R}(\mathcal{K})\) et \(\mathbb{R}(\mathcal{L})\), on constate que ces lattices sont isomorphes avec la lattice \(\mathbb{R}^s_f(\varepsilon\mathcal{L})\) des sous-catégories \(\mathcal{L}\)-semi-réflexives de la catégorie \(\mathcal{C}_2\mathcal{V}\), et l'isomorphisme des lattices \(\mathbb{K}(\mathcal{K})\) et \(\mathbb{K}(\mathcal{L})\) nous mène à leur isomorphisme avec la lattice \(\mathbb{K}^s_f(\mu\mathcal{K})\) des sous-catégories \(\mathcal{K}\)-semi-coréflexives.A note on cores and quasi relative interiors in partially finite convex programminghttps://zbmath.org/1496.460772022-11-17T18:59:28.764376Z"Lindstrom, Scott B."https://zbmath.org/authors/?q=ai:lindstrom.scott-bSummary: The problem of minimizing an entropy functional subject to linear constraints is a useful example of partially finite convex programming. In the 1990s, Borwein and Lewis provided broad and easy-to-verify conditions that guarantee strong duality for such problems. Their approach is to construct a function in the quasi-relative interior of the relevant infinite-dimensional set, which assures the existence of a point in the core of the relevant finite-dimensional set. We revisit this problem, and provide an alternative proof by directly appealing to the definition of the core, rather than by relying on any properties of the quasi-relative interior. Our approach admits a minor relaxation of the linear independence requirements in Borwein and Lewis' framework, which allows us to work with certain piecewise-defined moment functions precluded by their conditions. We provide such a computed example that illustrates how this relaxation may be used to tame observed Gibbs phenomenon when the underlying data is discontinuous. The relaxation illustrates the understanding we may gain by tackling partially-finite problems from both the finite-dimensional and infinite-dimensional sides. The comparison of these two approaches is informative, as both proofs are constructive.Hyperstability of orthogonally 3-Lie homomorphism: an orthogonally fixed point approachhttps://zbmath.org/1496.460782022-11-17T18:59:28.764376Z"Keshavarz, Vahid"https://zbmath.org/authors/?q=ai:keshavarz.vahid"Jahedi, Sedigheh"https://zbmath.org/authors/?q=ai:jahedi.sedighehSummary: In this chapter, by using the orthogonally fixed point method, we prove the Hyers-Ulam stability and the hyperstability of orthogonally 3-Lie homomorphisms for additive \(\rho\)-functional equation in 3-Lie algebras. Indeed, we investigate the stability and the hyperstability of the system of functional equations
\[
\begin{cases}
f(x+y)-f(x)-f(y)= \rho (2f(\frac{x+y}{2})+ f(x)+ f(y)),\\
f([[u,v],w])=[[f(u),f(v)],f(w)]
\end{cases}
\]
in 3-Lie algebras where \(\rho \neq 1\) is a fixed real number.
For the entire collection see [Zbl 1485.65002].Fuzzy bounded operators with application to Radon transformhttps://zbmath.org/1496.460792022-11-17T18:59:28.764376Z"Bînzar, Tudor"https://zbmath.org/authors/?q=ai:binzar.tudor"Pater, Flavius"https://zbmath.org/authors/?q=ai:pater.flavius-lucian"Nădăban, Sorin"https://zbmath.org/authors/?q=ai:nadaban.sorin-florinSummary: This paper is focused on developing the means to extend the range of application of the inverse Radon transform by enlarging the domain of definition of the Radon operator, namely from a specific Banach space to a more general fuzzy normed linear space. This is done by studying different types of fuzzy bounded linear operators acting between fuzzy normed linear spaces. The motivation for considering this type of spaces comes from the existence of an equivalence between the probabilistic metric spaces and fuzzy metric spaces, in particular fuzzy normed linear spaces. We mention that many notions and results belonging to classical metric spaces could also be found in this general context. Moreover, this setup allows to develop applications as diverse as: image processing, data compression, signal processing, computer graphics, etc. The class of operators that best fits the intended purpose is the class of strongly fuzzy bounded linear operators. The main results about this family of operators use the fact that the space of such operators becomes a normed algebra. An extension of the classical norm of a bounded linear operator between two normed spaces to the norm of strongly fuzzy bounded linear operators acting between fuzzy normed linear spaces is proved. A version of the classical Banach-Steinhaus theorem for strongly fuzzy bounded linear operators is given. A sufficient condition for the limit of a sequence of strongly fuzzy bounded linear operators to be strongly fuzzy bounded is shown. The adjoint operator of a strongly fuzzy bounded linear operator is a classic bounded linear operator. The class of neighborhood fuzzy bounded linear operators are studied as well, being established connections with two other classes of operator, namely the class of fuzzy bounded linear operators and strongly fuzzy bounded linear operators.Fuzzy-interval inequalities for generalized preinvex fuzzy interval valued functionshttps://zbmath.org/1496.460802022-11-17T18:59:28.764376Z"Khan, Muhammad Bilal"https://zbmath.org/authors/?q=ai:bilal-khan.muhammad"Srivastava, Hari Mohan"https://zbmath.org/authors/?q=ai:srivastava.hari-mohan"Mohammed, Pshtiwan Othman"https://zbmath.org/authors/?q=ai:mohammed.pshtiwan-othman"Guirao, Juan L. G."https://zbmath.org/authors/?q=ai:garcia-guirao.juan-luis"Jawa, Taghreed M."https://zbmath.org/authors/?q=ai:jawa.taghreed-mSummary: In this paper, firstly we define the concept of \(h\)-preinvex fuzzy-interval-valued functions (\(h\)-preinvex FIVF). Secondly, some new Hermite-Hadamard type inequalities (\(H\)-\(H\) type inequalities) for \(h\)-preinvex FIVFs via fuzzy integrals are established by means of fuzzy order relation. Finally, we obtain Hermite-Hadamard Fejér type inequalities (\(H\)-\(H\) Fejér type inequalities) for \(h\)-preinvex FIVFs by using above relationship. To strengthen our result, we provide some examples to illustrate the validation of our results, and several new and previously known results are obtained.Almost invariant subspaces of the shift operator on vector-valued Hardy spaceshttps://zbmath.org/1496.470192022-11-17T18:59:28.764376Z"Chattopadhyay, Arup"https://zbmath.org/authors/?q=ai:chattopadhyay.arup"Das, Soma"https://zbmath.org/authors/?q=ai:das.soma"Pradhan, Chandan"https://zbmath.org/authors/?q=ai:pradhan.chandanThe authors characterize nearly invariant subspaces of finite defect for the backward shift operator acting on vector-valued Hardy spaces \(H^2_{\mathbb C^m}(\mathbb D)\), generalizing the scalar-valued result by \textit{I. Chalendar} et al. [J. Oper. Theory 83, No. 2, 321--331 (2020; Zbl 1463.47096)]. Given a bounded analytic function \(\Theta\) with values in the space of linear operators \(\mathcal L(\mathbb C^r,\mathbb C^m)\), we can induce the multiplier \(T_\theta F(z)=\Theta(z)F(z)\) from \(H^2_{\mathbb C^r}(\mathbb D)\) into \(H^2_{\mathbb C^m}(\mathbb D)\). They are determined by the condition \(ST_\Theta=T_\Theta S\) where \(S\) denotes the forward shift operator \(SF(z)=zF(z)\) which acts on the corresponding space in each case. We write, as usual, the backward shift \(S^* F(z)=\frac{F(z)-F(0)}{z}\). A closed subspace \(\mathcal M\subset H^2_{\mathbb C^m}(\mathbb D)\) is called almost-invariant for \(S\) if there exists a finite-dimensional subspace \(\mathcal F\subset H^2_{\mathbb C^m}(\mathbb D)\) such that \(S(\mathcal M)\subset \mathcal M\oplus\mathcal F\). Similarly, a closed subspace \(\mathcal M\subset H^2_{\mathbb C^m}(\mathbb D)\) is called nearly invariant for \(S^*\) if any \(F\in \mathcal M\) with \(F(0)=0\) satisfies that \(S^*F\in \mathcal M\) and it is called nearly \(S^*\)-invariant with defect \(p\) if there exists a \(p\)-dimensional subspace \(\mathcal F\subset H^2_{\mathbb C^m}(\mathbb D)\) such that, if \(F\in \mathcal M\) with \(F(0)=0\), then \(S^*F\in \mathcal M \oplus \mathcal F\). Also, \(\mathcal M\) is called \(S^*\)-almost invariant with defect \(p\) if \(S^*\mathcal M\subset\mathcal M \oplus \mathcal F\) and \(\dim \mathcal F=p\).
In the paper under review, the authors present a characterization of nearly invariant subspaces for \(S^*\) with finite defect in the vector-valued Hardy spaces. Using such a result, they also manage to obtain the description of almost invariant subspaces for the shift and its adjoint acting on vector-valued Hardy spaces.
Reviewer: Oscar Blasco (València)Equivalence of semi-norms related to super weakly compact operatorshttps://zbmath.org/1496.470212022-11-17T18:59:28.764376Z"Tu, Kun"https://zbmath.org/authors/?q=ai:tu.kunLet $X$ and $Y$ be real infinite-dimensional Banach spaces. A subset $A$ of $X$ is said to be relatively super weakly compact if $A_{\mathcal{U}}$ is relatively weakly compact in $X_{\mathcal{U}}$ for any free ultrafilter $\mathcal{U}$. $A$ is said to be super weakly compact if it is weakly closed and relatively super weakly compact. The measure of super weak noncompactness of a bounded subset $A$ of $X$, $\sigma(A)$ is defined as
\[
\sigma(A)=\inf\{t > 0 : A\subset S + tB_X,\ S\text{ is relatively super weakly compact}\}.
\]
A bounded linear operator $T:X\to Y$ is called super weakly compact if $T(B_X)$ is relatively super weakly compact. Equivalently, $T_{\mathcal{U}}$ is weakly compact for any free ultrafilter $\mathcal{U}$. $A$ is called weakly compact if $T(B_X)$ is relatively weakly compact where $B_X$ is the closed unit ball in $X$.
Let $L(X,Y)$ denote the collection of all bounded linear operators mapping $X$ to $Y$ and $S(X,Y)$ represent the collection of all super weakly compact operators. The super weak essential norm $\|\cdot\|_s$ of $T\in L(X,Y)$ is the semi-norm induced from the quotient space $L(X,Y)/S(X,Y)$, that is,
\[
\|T\|_s=\inf\{\|T -S\|:S\in S(X,Y)\}.
\]
The space $X$ is said to have the super weakly compact approximation property (SWAP) if there is a real number $\lambda>0$ such that for any super weakly compact set $A\subset X$ and any $\varepsilon>0$, there is a super weakly compact operator $R:X\to X$ with $\sup_{x\in A} \|x- Rx\|\leq\varepsilon$ and $\|R\|\leq\lambda$.
In this paper, super weakly compact operators are discussed through a quantative method. By introducing the semi-norm $\sigma(T)$ of the operator $T:X\to Y$, which measures how far $T$ is from the family of super weakly compact operators, the following equivalence of the measure $\sigma(T)$ and the super weak essential norm $\|T\|_s$ of $T$ is proved:
Theorem. A Banach space $Y$ has the (SWAP) if only if the semi-norms $\sigma$ and ${\| \cdot\|}_s$ are equivalent in $L(X,Y)$ for any Banach space $X$.
In order to give an application of this theorem, some basic properties of Banach spaces having the SWAP are studied in Section~4 of the paper and then an example is constructed to show that the measures of $T$ and its dual $T^*$ are not always equivalent. Moreover, some examples of Banach spaces which have and which do not have the SWAP are given in this paper.
Reviewer: T. D. Narang (Amritsar)Subexponential decay and regularity estimates for eigenfunctions of localization operatorshttps://zbmath.org/1496.470352022-11-17T18:59:28.764376Z"Bastianoni, Federico"https://zbmath.org/authors/?q=ai:bastianoni.federico"Teofanov, Nenad"https://zbmath.org/authors/?q=ai:teofanov.nenadInspired by the recent work [\textit{D. Bayer} and \textit{K. Gröchenig}, Integral Equations Oper. Theory 82, No. 1, 95--117 (2015; Zbl 1337.47029)], the authors consider the properties of eigenfunctions of compact localization operators. They extend the framework of the Schwartz space of test functions and its dual space of tempered distributions given in the previous reference by replacing it with a more subtle family of Gelfand-Shilov spaces and their duals, spaces of ultra-distributions. As an important technical tool, the authors consider a class of weights which contains the weights of subexponential growth, apart from polynomial type weights. Among the main tools is the \(\tau \)-Wigner distribution \(W_{\tau }(f,g)\) with \(f,g\in L^{2}(\mathbb{R}^{2}).\)
Reviewer: Elhadj Dahia (Bou Saâda)Extrapolation theorems for \((p,q)\)-factorable operatorshttps://zbmath.org/1496.470362022-11-17T18:59:28.764376Z"Galdames-Bravo, Orlando"https://zbmath.org/authors/?q=ai:galdames-bravo.orlandoSummary: The operator ideal of \((p,q)\)-factorable operators can be characterized as the class of operators that factors through the embedding \(L^{q'}(\mu)\hookrightarrow L^{p}(\mu)\) for a finite measure \(\mu\), where \(p,q\in[1,\infty)\) are such that \(1/p+1/q\geq1\). We prove that this operator ideal is included into a Banach operator ideal characterized by means of factorizations through \(r\)th and \(s\)th power factorable operators, for suitable \(r,s\in[1,\infty)\). Thus, they also factor through a positive map \(L^{s}(m_{1})^{\ast}\to L^{r}(m_{2})\), where \(m_{1}\) and \(m_{2}\) are vector measures. We use the properties of the spaces of \(u\)-integrable functions with respect to a vector measure and the \(u\)th power factorable operators to obtain a characterization of \((p,q)\)-factorable operators and conditions under which a \((p,q)\)-factorable operator is \(r\)-summing for \(r\in[1,p]\).A class of \(C\)-normal weighted composition operators on Fock space \(\mathcal{F}^2 (\mathbb{C})\)https://zbmath.org/1496.470462022-11-17T18:59:28.764376Z"Bhuia, Sudip Ranjan"https://zbmath.org/authors/?q=ai:bhuia.sudip-ranjanSummary: In this paper we study a class of \(C\)-normal weighted composition operators \(W_{\psi, \varphi}\) on the Fock space \(\mathcal{F}^2 (\mathbb{C})\). We provide some properties of \(\psi\) and \(\varphi\) when a weighted composition operator \(W_{\psi, \varphi}\) is \(C\)-normal with a conjugation \(C\) defined on \(\mathcal{F}^2 (\mathbb{C})\). We also show that hyponormal \(C\)-normal operators are normal. We investigate \(C\)-normality of \(W_{\psi, \varphi}\) with weighted composition conjugation and the weight function as a kernel function. Alongside we give eigenvalues and eigenvectors of \(C\)-normal \(W_{\psi, \varphi}\). We establish a condition on the symbols for a class of \(C\)-normal weighted composition operators to be normal.On double difference of composition operators from a space generated by the Cauchy kernel and a special measurehttps://zbmath.org/1496.470482022-11-17T18:59:28.764376Z"Sharma, Mehak"https://zbmath.org/authors/?q=ai:sharma.mehak"Sharma, Ajay K."https://zbmath.org/authors/?q=ai:sharma.ajay-kumar|sharma.ayay-k"Mursaleen, M."https://zbmath.org/authors/?q=ai:mursaleen.mohammad-ayman|mursaleen.mohammadSummary: In this paper, compact double difference of composition operators acting from a space generated by the Cauchy kernel and a special measure to analytic Besov spaces is characterized. Moreover, operator norm of these operators acting from Cauchy transforms to analytic Besov spaces is obtained explicitly.Reflexive operators on analytic function spaceshttps://zbmath.org/1496.470592022-11-17T18:59:28.764376Z"Ershad, F."https://zbmath.org/authors/?q=ai:ershad.fariba"Khorami, M. M."https://zbmath.org/authors/?q=ai:khorami.mohammad-morad"Yousefi, B."https://zbmath.org/authors/?q=ai:yousefi.bahmannSummary: Following the recent work done in [\textit{B. Yousefi} and \textit{F. Zangeneh}, Proc. Indian Acad. Sci., Math. Sci. 128, No. 3, Paper No. 38, 9 p. (2018; Zbl 06911924)], we give various other conditions to ensure that the powers of the multiplication operator \(M_z\) are reflexive on a Banach space \({\mathcal{X}}\) of functions analytic on a plane domain. Also, some examples of function spaces satisfying the given conditions are considered.Compactness of multiplication operators on Riesz bounded variation spaceshttps://zbmath.org/1496.470602022-11-17T18:59:28.764376Z"Guzmán-Partida, Martha"https://zbmath.org/authors/?q=ai:guzman-partida.marthaSummary: We prove compactness of the operator \(M_h C_g\) on a subspace of the space of \(2 \pi\)-periodic functions of Riesz bounded variation on \([-\pi,\pi]\), for appropriate functions \(g\) and \(h\). Here, \(M_h\) denotes multiplication by \(h\) and \(C_g\) convolution by \(g\).Grüss-Landau inequalities for elementary operators and inner product type transformers in \(\mathrm{Q}\) and \(\mathrm{Q}^*\) norm ideals of compact operatorshttps://zbmath.org/1496.470622022-11-17T18:59:28.764376Z"Lazarević, Milan"https://zbmath.org/authors/?q=ai:lazarevic.milanSummary: For a probability measure $\mu$ on $\Omega$ and square integrable (Hilbert space) operator valued functions $\{A^*_t\}_{t\in \Omega}$, $\{B_t\}_{t\in\Omega}$, we prove Grüss-Landau type operator inequality for inner product type transformers
$$
\begin{multlined}
\left| \int_\Omega A_t X B_t \,d\mu(t) - \int_\Omega A_t\,d\mu(t) X \int_\Omega B_t \,d\mu(t) \right|^{2\eta} \\
\leqslant
\left\Vert \int_\Omega A_t A^*_t\,d\mu(t) - \left| \int_\Omega A^*_t \,d\mu(t) \right|^2 \right\Vert^\eta \left( \int_\Omega B^*_t X^* X B_t \,d\mu(t) - \left| X \int_\Omega B_t \,d\mu(t)\right|^2 \right)^\eta,
\end{multlined}
$$
for all $X \in \mathcal{B(H)}$ and for all $\eta \in [0,1]$.
Let $p\geqslant2$, $\Phi$ to be a symmetrically norming (s.n.) function, $\Phi^{(p)}$ to be its $p$-modification, $\Phi^{(p)^*}$ is a s.n. function adjoint to $\Phi^{(p)}$ and $\Vert\cdot\Vert_{\Phi^{(p)^*}}$ to be a norm on its associated ideal $\mathcal{C}_{\Phi^{(p)^*}}(\mathcal{H})$ of compact operators. If $X\in \mathcal{C}_{\Phi^{(p)^*}}(\mathcal{H})$ and $\{\alpha_n\}^\infty_{n=1}$ is a sequence in $(0,1]$, such that $\sum^\infty_{n=1}\alpha_n=1$ and $\sum^\infty_{n=1}\Vert\alpha ^{-1/2}_n A_n f \Vert^2 + \Vert \alpha^{-1/2}_n B^*_n f \Vert^2<+\infty$ for some families $\{A_n\}^\infty_{n=1}$ and $\{B_n\}^\infty_{n=1}$ of bounded operators on Hilbert space $\mathcal{H}$ and for all $f\in \mathcal{H}$, then
$$
\begin{multlined}
\left\Vert\sum^\infty_{n=1} \alpha^{-1}_n A_n X B_n - \sum^\infty_{n=1} A_n X \sum^\infty_{n=1} B_n \right\Vert _{\Phi^{(p)^*}} \\
\leqslant
\left\Vert \sqrt{ \sum^\infty_{n=1} \alpha^{-1}_n |A_n|^2 - \left| \sum^\infty_{n=1} A_n \right|^2} X \sqrt{ \sum^\infty_{n=1} \alpha^{-1}_n |B^*_n|^2 - \left| \sum^\infty_{n=1} B^*_n\right|^2} \right\Vert_{\Phi^{(p)^*}},
\end{multlined}
$$
if at least one of those operator families consists of mutually commuting normal operators.
The related Grüss-Landau type $\Vert\cdot\Vert_{\Phi^{(p)}}$ norm inequalities for inner product type transformers are also provided.Spectrally additive group homomorphisms on Banach algebrashttps://zbmath.org/1496.470632022-11-17T18:59:28.764376Z"Askes, Miles"https://zbmath.org/authors/?q=ai:askes.miles"Brits, Rudi"https://zbmath.org/authors/?q=ai:brits.rudi-m"Schulz, Francois"https://zbmath.org/authors/?q=ai:schulz.francoisSummary: Let \(A\) and \(B\) be unital complex Banach algebras. A surjective map \(\phi : A \to B\) is called a spectrally additive group homomorphism if the spectrum of \(x \pm y\) is equal to the spectrum of \(\phi (x) \pm \phi (y)\) for each \(x, y \in A\). If \(A\) is semisimple and either \(A\) or \(B\) has an essential socle, then we prove that a spectrally additive group homomorphism \(\phi : A \to B\) is a continuous Jordan-isomorphism. If, in addition, either \(A\) or \(B\) is prime, then we conclude that \(\phi\) is either a continuous algebra isomorphism or anti-isomorphism. It is noteworthy that the continuity and linearity (or even additivity) of the map \(\phi\) does not form part of the hypothesis, but is rather obtained in the conclusion. The techniques employed in the proof of these results utilize the spectral rank, trace and determinant, and yield a new additive characterization of finite rank elements in a Banach algebra which is of independent interest.The Kalton and Rosenthal type decomposition of operators in Köthe-Bochner spaceshttps://zbmath.org/1496.470652022-11-17T18:59:28.764376Z"Pliev, Marat"https://zbmath.org/authors/?q=ai:pliev.marat-amurkhanovich"Sukochev, Fedor"https://zbmath.org/authors/?q=ai:sukochev.fedor-aThe paper under review is partly motivated by a classical result of \textit{N. J. Kalton} [Indiana Univ. Math. J. 27, 353--381 (1978; Zbl 0403.46032)] which in particular implies that every operator \(T:L_1(\mu)\rightarrow L_1(\mu)\) admits a decomposition \(T=T_a+T_d\) where \(T_a\) is a pseudo-integral operator with respect to a family of absolutely continuous measures and \(T_d\) is an atomic operator. This decomposition is here extended to a certain class of operators (dominated operators) from a certain family of lattice-normed spaces (decomposable spaces) into order continuous Banach lattices. Here, a lattice-normed space is a triple \((E,V,p)\), where \(E\) is a vector space, \(V\) is an Archimedean vector lattice, and \(p:E\rightarrow V\) is a map satisfying: (1) \(p(x)\geq0\), with \(p(x)=0\) if and only if \(x=0\); (2) \(p(x_1+x_2)\leq p(x_1)+p(x_2)\); (3) \(p(\lambda x)=|\lambda|p(x)\). In this setting, several interesting results are obtained concerning the class of disjointness preserving and narrow operators. Moreover, the results are in particular applied to the situation of operators on Köthe-Bochner spaces. We refer the interested reader to the paper for details.
Reviewer: Pedro Tradacete (Madrid)Global right inverses for Euler type differential operators on the space of smooth functionshttps://zbmath.org/1496.470712022-11-17T18:59:28.764376Z"Langenbruch, Michael"https://zbmath.org/authors/?q=ai:langenbruch.michaelSummary: We study Euler type partial differential operators \(P(\theta)\) admitting a continuous linear right inverse on the space of smooth functions defined on \(\mathbb{R}^d\). Specifically, if the canonical unit vectors \(\{ e_j | j \leq d \}\) are non characteristic for \(P\) then \(P(\theta)\) admits a continuous linear right inverse iff \(P(\partial)\) is hyperbolic w.r.t.\ \( e_j\) for any \(j \leq d\).New \(L^p\)-inequalities for hyperbolic weights concerning the operators with complex Gaussian kernelshttps://zbmath.org/1496.470732022-11-17T18:59:28.764376Z"González, Benito J."https://zbmath.org/authors/?q=ai:gonzalez.benito-juan"Negrín, Emilio R."https://zbmath.org/authors/?q=ai:negrin.emilio-rSummary: In this article, the authors present a systematic study of several new \(L^{p}\)-boundedness properties and Parseval-type relations concerning the operators with complex Gaussian kernels over the spaces \(L^{p}(\mathbb{R},\cosh(\alpha x)\,dx)\) and \(L^{p}(\mathbb{R},\cosh(\alpha x^{2})\,dx)\), \(1\leq p\leq\infty\), \(\alpha\in\mathbb{R}\). Relevant connections with various earlier related results are also pointed out.Erdélyi-Kober fractional integral operators on ball Banach function spaceshttps://zbmath.org/1496.470742022-11-17T18:59:28.764376Z"Ho, Kwok-Pun"https://zbmath.org/authors/?q=ai:ho.kwok-punThe author studies ball Banach function spaces and the Erdélyi-Kober fractional integral operators. The boundedness of the above operators on ball Banach function spaces is derived and proved. Furthermore, the boundedness of Erdélyi-Kober fractional integral operators on amalgan and Morrey spaces is also derived and proved.
Reviewer: James Adedayo Oguntuase (Abeokuta)Characterizations of pseudo-differential operators on \(\mathbb{S}^1\) based on separation-preserving operatorshttps://zbmath.org/1496.470752022-11-17T18:59:28.764376Z"Faghih, Zahra"https://zbmath.org/authors/?q=ai:faghih.zahra"Ghaemi, M. B."https://zbmath.org/authors/?q=ai:ghaemi.mohammad-bagherSummary: In this paper, we prove that a bounded pseudo-differential operator \(T_{\sigma}:L^p(\mathbb{S}^1)\rightarrow L^p(\mathbb{S}^1)\) for \(1\leq p<\infty\), is a separation-preserving operator and give a formula for its symbols \(\sigma\). Using these formulas, we give a new representation for the symbol of adjoint and products of two pseudo-differential operators.Cohen summing multilinear multiplication operatorshttps://zbmath.org/1496.470862022-11-17T18:59:28.764376Z"Popa, Dumitru"https://zbmath.org/authors/?q=ai:popa.dumitruA continuous linear operator between Banach spaces \(T\in\mathcal L(X,Y)\) is strongly \(p\)-summing in the sense of \textit{J. S. Cohen} [Math. Ann. 201, 177--200 (1973; Zbl 0233.47019)] if and only if its adjoint \(T^*\in\mathcal L(Y^*,X^*)\) is absolutely \(p\)-summing. As an easy consequence of a classical result about absolutely \(p\)-summing maps with range on cotype 2 spaces, the following statement is derived: If \(X\) has type 2, then, for every \(1<p<2\), any strongly \(p\)-summing operator \(T\in\mathcal L(X,Y)\) is strongly 2-summing.
The notion of strongly \(p\)-summability in the sense of Cohen has its multilinear counterpart. A bounded \(k\)-linear operator \(U:X_1\times\cdots\times X_k\to Y\) is \textit{Cohen} \(p\)-\textit{summing} if the linear adjoint \(U^*\in\mathcal L(Y^*, \mathcal L(X_1,\dots, X_k))\) is absolutely \(p\)-summing. \textit{Q. Bu} and \textit{Z. Shi} [J. Math. Anal. Appl. 401, No. 1, 174--181 (2013; Zbl 1275.47117)] commented that they did not know whether the multilinear version of the above result holds. That is, if \(X_1,\dots, X_k\) have type \(p\), is it true that every Cohen \(p\)-summing operator from \(X_1\times\cdots\times X_k\) to \(Y\) is Cohen 2-summing, for every \(1<p<2\)?
Motivated by this question, the author of the paper under review examines Cohen \(p\)-summability for multilinear \textsl{multiplication} operators. As a result of this study, several examples are presented showing that the answear to Bu and Shi's question is negative.
Reviewer: Verónica Dimant (Victoria)The SHAI property for the operators on \(L^p\)https://zbmath.org/1496.471272022-11-17T18:59:28.764376Z"Johnson, W. B."https://zbmath.org/authors/?q=ai:johnson.william-b"Phillips, N. C."https://zbmath.org/authors/?q=ai:phillips.nicholas-c|phillips.n-c-k|phillips.n-christopher"Schechtman, G."https://zbmath.org/authors/?q=ai:schechtman.gideonThis work is motivated by the problem mentioned by \textit{B. Horváth} [Stud. Math. 253, No. 3, 259--282 (2020; Zbl 1460.46036)] whether \(L_{p}(0,1)\) (\(1<p<\infty\)) has the SHAI (surjective homomorphisms are injective) property provided that for every Banach space \(Y\), every continuous surjective algebra homomorphism from the bounded linear operators on \(L_{p}(0,1)\) onto the bounded linear operators on \(Y\) is injective. This problem is solved in the present paper (Corollary 1.6).
Reviewer: Elhadj Dahia (Bou Saâda)Reinhardt free spectrahedrahttps://zbmath.org/1496.471282022-11-17T18:59:28.764376Z"McCullough, Scott"https://zbmath.org/authors/?q=ai:mccullough.scott-a"Tuovila, Nicole"https://zbmath.org/authors/?q=ai:tuovila.nicoleSummary: Free spectrahedra are natural objects in the theories of operator systems and spaces and completely positive maps. They also appear in various engineering applications. In this paper, free spectrahedra satisfying a Reinhardt symmetry condition are characterized graph theoretically. It is also shown that, for a simple class of such spectrahedra, automorphisms are trivial.On the \(k\) point density problem for band-diagonal \(M\)-baseshttps://zbmath.org/1496.471292022-11-17T18:59:28.764376Z"Pyshkin, Alexey"https://zbmath.org/authors/?q=ai:pyshkin.alexeyLet \(\mathcal{H}\) be a real infinite-dimensional Hilbert space having a sequence of vectors \(\{f_n\}\) which is complete. Consider the operator algebra \(\mathcal{A}=\{T\in \mathcal{B}(\mathcal{H})\mid Tf_n=\lambda_nf_n,\,\lambda_n\in \mathbb{R}\}\) and the algebra \(R_1(\mathcal{A})\) generated by rank one operators of \(\mathcal{A}\). The algebra \(R_1(A)\) is said to be \(k\) point dense in \(\mathcal{A}\) if for any \(x_1,x_2,\dots,x_k\in \mathcal{H}\) and \(\epsilon>0\), there exists \(R\in R_1(\mathcal{A})\) such that \(\Vert Rx_s-x_s\Vert<\epsilon\) for any \(1\leq s\leq k\). The algebra \(\mathcal{A}\) has rank one density property if the unit ball of \(R_1(\mathcal{A})\) is dense in the unit ball of \(\mathcal{A}\) in the strong operator topology. In [\textit{W. E. Longstaff}, Can. J. Math. 28, 19--23 (1976; Zbl 0317.46052)], an interesting question was raised: does one point density property imply rank one density property? The answer was given in the negative in [\textit{D. R. Larson} and \textit{W. R. Wogen}, J. Funct. Anal. 92, No. 2, 448--467 (1990; Zbl 0738.47045)].
The author considers band-diagonal systems similar to the one regarded by Larson and Wogen [loc.\,cit.]\ to determine the exact conditions for \(k\) point density property of such vector systems. Simpler proofs of results in Larson and Wogen's paper [loc.\,cit.] are provided. Finally, a similar theorem for a pentadiagonal system is proved.
Reviewer: Ajay Kumar (Delhi)Gaffney-Friedrichs inequality for differential forms on Heisenberg groupshttps://zbmath.org/1496.490222022-11-17T18:59:28.764376Z"Franchi, Bruno"https://zbmath.org/authors/?q=ai:franchi.bruno"Montefalcone, Francescopaolo"https://zbmath.org/authors/?q=ai:montefalcone.francescopaolo"Serra, Elena"https://zbmath.org/authors/?q=ai:serra.elenaSummary: In this paper, we will prove several generalized versions, dependent on different boundary conditions, of the classical Gaffney-Friedrichs inequality for differential forms on Heisenberg groups. In the first part of the paper, we will consider horizontal differential forms and the horizontal differential. In the second part, we shall prove the counterpart of these results in the context of Rumin's complex.Asymptotic geometric analysis. Part IIhttps://zbmath.org/1496.520012022-11-17T18:59:28.764376Z"Artstein-Avidan, Shiri"https://zbmath.org/authors/?q=ai:artstein-avidan.shiri"Giannopoulos, Apostolos"https://zbmath.org/authors/?q=ai:giannopoulos.apostolos-a"Milman, Vitali D."https://zbmath.org/authors/?q=ai:milman.vitali-dThis book is the continuation of the excellent monograph [\textit{S. Artstein-Avidan} et al., Asymptotic geometric analysis. I. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1337.52001)]. In this series of two books the modern theory of Asymptotic Geometric Analysis is presented. This theory had its origin in the field of (infinite dimensional) Functional Analysis, and evolved largely to a finite dimensional theory, but where the dimension of the objects of study (normed spaces, convex bodies, convex functions...) is very high, increasing to infinity. Both monographs are outstanding and essential references for the study of this theory.
The first four chapters are the natural continuation of the first volume. Thus, in Chapter 1 the authors revisit and deepen the concentration of measure phenomenon (cf. Chapter 3 of Part I), including its relation with important functional inequalities: First, Poincaré's inequality and its role in concentration is studied, and a proof of it in the Gaussian space based on the Ornstein-Uhlenbeck semigroup is provided; also the concentration on the discrete cube is exhaustively analyzed. Then, cost-induced transforms are discussed, including inequalities such as (Weak) Cost-Santaló inequality. I would like to mention another major inequality connected to this question of concentration, the logarithmic Sobolev inequality, which is also studied thoroughly.
Chapter 2 is devoted to discussing major results and problems on isotropic log-concave probability measures, being a follow-up of Chapter 10 in Part I. Among others, the authors deal with the famous Kannan-Lovász-Simonovits conjecture, its equivalent formulation saying that Poincaré's inequality holds for every isotropic log-concave probability measure on \(\mathbb{R}^n\) with a constant independent of the measure or the dimension, or the central limit problem. Important and breakthrough works on these questions by e.g. E. Milman, Eldan, Klartag or Chen are described.
The Gaussian distribution is one of the cornerstones in probability theory, and Chapter 3 deals with some fundamental isoperimetric inequalities about the \(n\)-dimensional Gaussian measure \(\gamma_n\) (cf. Chapter 9 of Part I). Naturally, one of the main results in this chapter is the engaging form of the isoperimetric inequality stating that if \(A\) is a Borel set in \(\mathbb{R}^n\) and \(H\) is a half-space with \(\gamma_n(A)=\gamma_n(H)\), then \(\gamma_n(A_t)\geq\gamma_n(H_t)\) for all \(t>0\), being \(A_t\) the \(t\)-extension of \(A\). Three different proofs of this important fact are presented (based on reducing it to the isoperimetric problem for the sphere, on a functional inequality or on a Gaussian symmetrization). Other significant results on this topic are studied here, such as Ehrhard's inequality, the behavior of the Gaussian measure with respect to dilates of a centrally symmetric convex body, the Gaussian correlation inequality, or the \(B\)-theorem of Cordero-Erausquin, Fradelizi and Maurey. Some applications of geometric inequalities for the Gaussian measure to discrepancy problems are also presented.
In Chapter 4, different volume-type inequalities are studied, continuing so the analysis that was made in Chapters 2 and 10 of Part I. First, the Brascamp-Lieb-Luttinger inequality, and the multidimensional versions of the Brascamp-Lieb and the Barthe inequalities are discussed, and several applications of these thorough results to different problems in Convex Geometry are presented. We highlight the Gluskin-Milman theorem, which is applied to show that every \(n\)-dimensional normed space has the random cotype-2 property. Next, volume estimates for convex bodies with few vertices or facets are exposed, as Vaaler's inequality bounding from below the volume of the intersection of a finite number of origin-symmetric strips, and other related results. Shephard's problem is also discussed, as well as its (strongly) negative answers by Petty and Schneider, and by Ball. Then the authors outline a theory that has been developed in several works of Paouris, Pivovarov et al., which provides a unified way of showing well-known inequalities from geometric probability. The chapter concludes considering the Blaschke-Petkantschin formulae and their geometric applications. Among them, Giannopoulos-Koldobsky's positive answer to a variant of the Busemann-Petty (and Shephard) problem, proposed by Milman, is presented.
Chapter 5 is devoted to the delightful theory of type and cotype, introduced and developed mainly by Maurey and Pisier. It starts with several basic notions and facts, and continues discussing the absolutely summing operators, nuclear operators, trace duality, the Gaussian type and cotype and the \(\ell\)-norm. Then, some developments on the duality of entropy problem for spaces with type \(p\) are analyzed, as well as some results for spaces with bounded cotype constant: the so-called ``Maurey-Pisier lemma'' and a theorem by Bourgain and Milman asserting that the volume ratio of the unit ball of an \(n\)-dimensional normed space is bounded by a function of the cotype 2 constant of the space. The last half of the chapter focuses on: Grothendieck's inequality; Kwapien's theorem asserting that the (Banach-Mazur) distance from a Banach space \(X\) to some Hilbert space is bounded from above by the product of the type 2 and the cotype 2 constants of \(X\); a theorem of Lindenstrauss and Tzafriri providing a positive answer to the complemented subspace problem; and Krivine's theorem and a counterpart/strengthening of it by Maurey and Pisier. A result of Johnson and Schechtman about embedding \(\ell^m_p\) into \(\ell^n_1\) closes the chapter.
Next, in Chapter 6 the geometry of the family of all normed (\(n\)-dimensional) spaces equipped with the Banach-Mazur distance (i.e., the so-called Banach-Mazur compactum) is investigated. The question of computing the diameter of the compactum is the starting point of the chapter, and Gluskin's theorem is shown: there exists an absolute constant \(c>0\) such that, for any \(n\in\mathbb{N}\), there exist two \(n\)-dimensional normed spaces with distance greater than \(cn\). Also the problem to estimate the diameter of the compactum in the non-symmetric case is discussed (the best known upper bound is due to Rudelson), for which the method of random orthogonal factorizations is previously introduced, as well as several applications of it. Next the authors consider Pełcynski's question about the asymptotic growth of the radius of the Banach-Mazur compactum with respect to \(\ell^n_{\infty}\); this problem is still open, the best known upper bound due to Giannopoulos being \(O(n^{5/6})\). The chapter ends with the study of several results from the local theory of normed spaces: Elton's theorem; a result by Milman-Wolfson about spaces with maximal distance to the Euclidean space; Alon-Milman's theorem on the existence of \(k\)-dimensional subespaces of a normed space with small distance to either \(\ell^k_2\) or \(\ell^k_{\infty}\); and the Schechtman theorem on the dependence on \(\varepsilon\) of the critical dimension in Dvoretzky theorem.
Symmetrizations of sets are one of the cornerstones in Convexity, and a major question in this matter is knowing how fast a sequence of symmetrizations approaches an Euclidean ball starting from an arbitrary convex body. Chapter 7 focuses on this issue, departing from two important symmetrizations: the Steiner and the Minkowski symmetrizations. While the first one has been most commonly studied in other monographs (although not from the point of view here considered), it is not the case for the second one, so conferring an extra incentive to the reading of this chapter. It is worthwhile to stress that in order to achieve this goal, the authors use tools from Asymptotic Geometric Analysis rather than the methods in classical Convex Geometry, which allow to show that the number of required symmetrizations in order to get ``close'' to an Euclidean ball is linear in the dimension. Works of Bourgain, Lindenstrauss and Milman, and mainly of Klartag, on this regard are presented, for both the Minkowski and the Steiner symmetrizations.
Next, Chapter 8 deals with the method of interlacing families of polynomials, focusing on its applications to Geometric Functional Analysis and Convex Geometry. It starts with a result of Batson, Spielman and Srivastava asserting, geometrically speaking, that a John decomposition of the identity can be approximated by a John sub-decomposition with suitable weights, involving a linear (in the dimension) number of terms. Then the interlacing polynomials are studied, and used afterwards for proving the restricted invertibility principle; other forms and generalizations of this theorem are also considered, one of which allows to show the proportional Dvoretzky-Rogers factorization theorem. The chapter ends with a full overview of the Kadison-Singer problem.
The book concludes with a wonderful Chapter 9, entitled ``Functionalization of Geometry'', where several sorts of functions on \(\mathbb{R}^n\) are studied from a geometric point of view: among them, of course, log-concave functions are the leading ones. Their importance in this context lies in the fact that they can be seen as an extension of (closed) convex sets, and particular operations between them as fundamental constructions in convex geometry; as an outcome, every geometric inequality has an analytic counterpart. Thus, after a first section devoted to log-concavity, functional duality is studied and, among others, it is proved that the Legendre transform is, up to linear variants, the only transform \(T\) defined on the set of convex functions on \(\mathbb{R}^n\) satisfying that \(T\circ T=Id\) and that \(\varphi\leq\psi\) if and only if \(T\varphi\geq T\psi\). The chapter ends showing functional versions of several fundamental geometric inequalities: Brunn-Minkowski, Uryshon, Blaschke-Santaló, reverse Brunn-Minkowski and Blaschke-Santaló, Rogers-Shephard, etc.
In line with the first volume of this series, all chapters are enriched with a final section, a collection of ``Notes and remarks'', where the main references regarding the content of the chapter can be found, as well as applications and many other problems related to its subject. The bibliography is large and exhaustive, consisting of almost 750 items, and including both, classical and very recent and seminal references.
This series of two books will most certainly be a fundamental source for the study and investigation in the field of Asymptotic Geometric Analysis.
Reviewer: Maria A. Hernández Cifre (Murcia)A unified approach to collectively maximal elements in abstract convex spaceshttps://zbmath.org/1496.520022022-11-17T18:59:28.764376Z"Park, Sehie"https://zbmath.org/authors/?q=ai:park.sehieThe author establishes a very KKM type theorem in abstract convex spaces from which he then obtains an abstract collectively maximal element theorem. Finally, the author shows that a large number of previous theorems on the existence of maximal element and of equilibrium can be derived from his result.
Reviewer: Mircea Balaj (Oradea)On a version of the slicing problem for the surface area of convex bodieshttps://zbmath.org/1496.520052022-11-17T18:59:28.764376Z"Brazitikos, Silouanos"https://zbmath.org/authors/?q=ai:brazitikos.silouanos"Liakopoulos, Dimitris-Marios"https://zbmath.org/authors/?q=ai:liakopoulos.dimitris-mariosSummary: We study the slicing inequality for the surface area instead of volume. This is the question whether there exists a constant \(\alpha_n\) depending (or not) on the dimension \(n\) so that
\[S(K)\leqslant \alpha_n|K|^{\frac{1}{n}}\max_{\xi \in S^{n-1}}S(K\cap \xi^{\perp }),\]
where \(S\) denotes surface area and \(|\cdot |\) denotes volume. For any fixed dimension we provide a negative answer to this question, as well as to a weaker version in which sections are replaced by projections onto hyperplanes. We also study the same problem for sections and projections of lower dimension and for all the quermassintegrals of a convex body. Starting from these questions, we also introduce a number of natural parameters relating volume and surface area, and provide optimal upper and lower bounds for them. Finally, we show that, in contrast to the previous negative results, a variant of the problem which arises naturally from the surface area version of the equivalence of the isomorphic Busemann-Petty problem with the slicing problem has an affirmative answer.The metric projections onto closed convex cones in a Hilbert spacehttps://zbmath.org/1496.520062022-11-17T18:59:28.764376Z"Qiu, Yanqi"https://zbmath.org/authors/?q=ai:qiu.yanqi"Wang, Zipeng"https://zbmath.org/authors/?q=ai:wang.zipengSummary: We study the metric projection onto the closed convex cone in a real Hilbert space \(\mathscr{H}\) generated by a sequence \(\mathcal{V} = \{v_n\}_{n=0}^\infty\). The first main result of this article provides a sufficient condition under which the closed convex cone generated by \(\mathcal{V}\) coincides with the following set:
\[
\mathcal{C}[[\mathcal{V}]]: = \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0, \text{ the series }\sum_{n=0}^\infty a_n v_n \text{ converges in } \mathscr{H}\bigg\}.
\]
Then, by adapting classical results on general convex cones, we give a useful description of the metric projection onto \(\mathcal{C}[[\mathcal{V}]]\). As an application, we obtain the best approximations of many concrete functions in \(L^2([-1,1])\) by polynomials with nonnegative coefficients.Symmetrized Talagrand inequalities on Euclidean spaceshttps://zbmath.org/1496.520132022-11-17T18:59:28.764376Z"Tsuji, Hiroshi"https://zbmath.org/authors/?q=ai:tsuji.hiroshi.1From the Introduction: The Talagrand inequality, which is also called the Talagrand transportation inequality, is as follows: If \(m = e^{-V}\, L_n\) is a probability measure on \(\mathbb{R}^n\) with \(\nabla^2V\ge k\) for some \(k>0\), and \(\mu\in P_2(\mathbb{R}^n)\), then \(W_2^2(\mu, m) \le \mathrm{Ent}_m(\mu)/k\) holds, where \(L_n\) is the Lebesgue measure on \(\mathbb{R}^n\), \(P_2(\mathbb{R}^n)\) is the set of all probability measures on \(\mathbb{R}^n\) with finite second moment, \(W_2\) is the Wasserstein distance, and \(\mathrm{Ent}_m\) is the relative entropy (or the Kullback-Leibler distance) with respect to \(m\). More generally, it is known that the Talagrand inequality holds on metric measure spaces with similar conditions to those above, and there are many studies on refinements of the Talagrand inequality and relations with logarithmic Sobolev inequalities and Poincaré inequalities [\textit{C. Villani}, Optimal transport. Old and new. Berlin: Springer (2009; Zbl 1156.53003)]. This paper is motivated by \textit{M. Fathi}'s following result [Electron. Commun. Probab. 23, Paper No. 81, 9 p. (2018; Zbl 1400.35008)]:
Theorem 1. Let \(\mu\), \(\nu\in P_2(\mathbb{R}^n)\). The following assertions hold:
\begin{itemize}
\item[(\textbf{1})] Let \(m = e^{-V}\, L_n\) be a probability measure on \(\mathbb{R}^n\) such that \(V\in C^{\infty}(\mathbb{R}^n)\) is even and \(k\)-convex for some \(k>0\). If \(\nu\) is symmetric (i.e., its density with respect to \(L_n\) is even), then it holds that \(1/2\, W_2^2(\mu, \nu) \le 1/k (\mathrm{Ent}_m(\mu) + \mathrm{Ent}_m(\nu))\).
\item[(\textbf{2})] Let \(m=\gamma_n\) be the \(n\)-dimensional standard Gaussian measure. If \(\int_{\mathbb{R}^n}x\, d\nu(x)=0\), then it holds that \(1/2\, W_2^2(\mu, \nu) \le \mathrm{Ent}_{\gamma_n}(\mu) + \mathrm{Ent}_{\gamma_n}(\nu))\).
\end{itemize}
Moreover, the equality holds in (\textbf{2}) if and only if there exist some positive definite symmetric matrix \(A\in \mathbb{R}^{n\times n}\) and some \(a\in \mathbb{R}\) such that \(\mu\) is the Gaussian measure whose center is \(a\) and covariance matrix is \(A\), and \(\nu\) is the Gaussian measure whose center is \(0\) and covariance matrix is \(A^{-1}\).
Note that Theorem 1 (\textbf{1}) does not include (\textbf{2}). When \(m=\nu\) in (\textbf{1}), we recover the classical Talagrand inequality, and hence Theorem 1 is a refinement of the classical Talagrand inequality provided \(V\) is even. Using Fathi's paper as reference, the author of the present paper calls this type of inequality a symmetrized Talagrand inequality. M. Fathi proved the symmetrized Talagrand inequality by using optimal transport theory and convex geometry. Moreover, M. Fathi pointed out that the symmetrized Talagrand inequality for Gaussian measures is related to the functional Blaschke-Santalo inequality, which is well known and important in convex geometry. This paper considers refinements and extensions of the symmetrized Talagrand inequality.
The present paper is organized as follows. Some fundamental notions from optimal transport theory and functional inequalities are introduced, including Talagrand inequalities. Section 3 gives another form of the symmetrized Talagrand inequality by a self-improvement of Fathi's result (Theorem 1 (\textbf{2})). The barycenter of a probability measure plays an important role in this section. In Section 4, the author proves the main theorems and apply them to prove the corresponding \(HWI\) inequalities (between entropy \(H\), Wasserstein distance \(W\) and Fisher information \(I\)), logarithmic Sobolev inequalities and Poincaré inequalities. Moreover, the author gives an alternative proof of Theorem 1 and an extension on the real line in Section 4.3. In the final section, the author describes an application of the result in the previous subsection to the Blaschke-Santalo inequality for log-concave probability measures.
Reviewer: Viktor Ohanyan (Erevan)Relaxed energies, defect measures, and minimal currentshttps://zbmath.org/1496.580042022-11-17T18:59:28.764376Z"Lin, Fang-Hua"https://zbmath.org/authors/?q=ai:lin.fang-huaA natural existence question for a continuous harmonic map with a suitably given Dirichlet boundary value or in a given homotopic class (a problem posed by R. Schoen) remains open. The author briefly describes several earlier studies concerning energy minimizing harmonic maps, and maps that minimize the so-called relaxed energy from \(\mathbb{R}^3\) into \(S^2\). Of particular interest is the partial regularity and properties of possible singularities of such maps. A sketch proof of a formula conjectured by \textit{H. Brezis} and \textit{P. Mironescu} [Sobolev maps to the circle. From the perspective of analysis, geometry, and topology. New York, NY: Birkhäuser (2021; Zbl 07332819)] is provided, concerning the relaxed \(k\)-energy for Sobolev maps from \(\mathbb{R}^n\) to \(S^k\), for \(k>1\).
For the entire collection see [Zbl 1491.46003].
Reviewer: Vladimir Balan (Bucureşti)On Sobolev rough pathshttps://zbmath.org/1496.601192022-11-17T18:59:28.764376Z"Liu, Chong"https://zbmath.org/authors/?q=ai:liu.chong"Prömel, David J."https://zbmath.org/authors/?q=ai:promel.david-j"Teichmann, Josef"https://zbmath.org/authors/?q=ai:teichmann.josefThe authors consider spaces of rough paths with Sobolev regularity and associated controlled rough differential equations. First they define the space of Sobolev rough paths using a fractional Sobolev norm. Typical estimates used for rough paths work for \(p\)-variation or related seminorms, but not for the fractional Sobolev norm presenting the main obstacle. In a suitable topology, the authors introduce the controlled rough paths of Sobolev type requiring a remainder term in the mixed Hölder-variation space. Then they show that solutions to the controlled rough differential equations driven by Sobolev rough paths also possess Sobolev regularity. Finally the authors prove local Lipschitz continuity of the Itô-Lyons map on the space of Sobolev rough paths with arbitrary low regularity with respect to the initial value, vector field and the driving signal.
Reviewer: Maria Gordina (Storrs)A reduced basis method for fractional diffusion operators. Ihttps://zbmath.org/1496.652162022-11-17T18:59:28.764376Z"Danczul, Tobias"https://zbmath.org/authors/?q=ai:danczul.tobias"Schöberl, Joachim"https://zbmath.org/authors/?q=ai:schoberl.joachimThe authors propose and analyze a reduced basis method to evaluate fractional norms and apply fractional powers of elliptic operators. Their approach relies on several independent evaluations of \((I-t_i^2 \Delta )^{-1}f\) which can be computed in parallel way. Exponential rates of convergence for the optimal choice of sampling points \(t_i\) are proved. Several numerical experiments are provided to confirm the theoretical results.
Reviewer: Lijun Yi (Shanghai)The reproducing kernel viewpoints of general Ritz-Galerkin approximationhttps://zbmath.org/1496.652252022-11-17T18:59:28.764376Z"Liu, Zhiyong"https://zbmath.org/authors/?q=ai:liu.zhiyong"Xu, Qiuyan"https://zbmath.org/authors/?q=ai:xu.qiuyanSummary: In this paper, we introduce the new concepts of strictly positive definite functional and reproducing kernel functional space. We establish a connection between the strictly positive definite functional and the Ritz-Galerkin approximation. We prove a posteriori error estimate of the finite element method based on the reproducing kernel viewpoints.Global three-dimensional solvability the axisimmetric Stefan problem for quasilinear equationhttps://zbmath.org/1496.800052022-11-17T18:59:28.764376Z"Podgaev, A. G."https://zbmath.org/authors/?q=ai:podgaev.aleksandr-grigorevich"Prudnikov, V. Ya."https://zbmath.org/authors/?q=ai:prudnikov.v-ya"Kulesh, T. D."https://zbmath.org/authors/?q=ai:kulesh.t-dSummary: We prove results related to the study of the solvability of a problem with an unknown boundary by compactness methods. Relative compactness theorems are used, which were obtained in previous publications, adapted to the study of problems like the Stefan problem with an unknown boundary.
In previous papers, for the equation considered here, we studied the initial-boundary problem in a non-cylindrical domain with a given curvilinear boundary of class \(W^1_2\) and the problem for which, under the condition on the unknown boundary, the coefficient latent specific heat of fusion (in contrast to the Stefan problem, considered given here) was an unknown quantity.
Therefore, in some places calculations will be omitted that almost completely coincide with those set out in the works listed below. The proposed technique can be applied in more general situations: more phase transition boundaries, or more complex nonlinearities.
As a result, global over time, the regular solvability of a single-phase axisymmetric Stefan problem for a quasilinear three-dimensional parabolic equation with unknown boundary from the class \(W^1_4\), is proved.Transitivity and homogeneity of orthosets and inner-product spaces over subfields of \(\mathbb{R}\)https://zbmath.org/1496.810242022-11-17T18:59:28.764376Z"Vetterlein, Thomas"https://zbmath.org/authors/?q=ai:vetterlein.thomasSummary: An orthoset (also called an orthogonality space) is a set \(X\) equipped with a symmetric and irreflexive binary relation \(\perp\), called the orthogonality relation. In quantum physics, orthosets play an elementary role. In particular, a Hilbert space gives rise to an orthoset in a canonical way and can be reconstructed from it. We investigate in this paper the question to which extent real Hilbert spaces can be characterised as orthosets possessing suitable types of symmetries. We establish that orthosets fulfilling a transitivity as well as a certain homogeneity property arise from (anisotropic) Hermitian spaces. Moreover, restricting considerations to divisible automorphisms, we narrow down the possibilities to positive definite quadratic spaces over an ordered field. We eventually show that, under the additional requirement that the action of these automorphisms is quasiprimitive, the scalar field embeds into \({{\mathbb{R}}} \).Quantum state recovery via environment-assisted measurement and weak measurementhttps://zbmath.org/1496.810272022-11-17T18:59:28.764376Z"Harraz, Sajede"https://zbmath.org/authors/?q=ai:harraz.sajede"Cong, Shuang"https://zbmath.org/authors/?q=ai:cong.shuang"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-joseSummary: In this paper, we propose a quantum state recovery scheme based on environment assisted measurement with weak measurements and flip operations. Before the decoherence channel the weak measurement and flip operators are applied to gain some information about the state of the system and transfer it to a more robust state. Then we utilize environment assisted measurement and post-flip operations to bring the system as close as possible to its initial state. We illustrate our scheme and compare it with a scheme based on environment assisted measurement and weak measurement reversal in the case of a decoherence channel. We show that the success probability of our proposed scheme is significantly improved for all initial states. The proposed scheme is applicable for recovery of \(N\)-qubit state from any type of decoherence with at least one invertible Kraus operator. Also, the explicit formula of total fidelity and success probability for recovery of \(N\)-qubit GHZ state are derived.Weak measurement effects on dynamics of quantum correlations in a two-atom system in thermal reservoirshttps://zbmath.org/1496.810342022-11-17T18:59:28.764376Z"Ananth, N."https://zbmath.org/authors/?q=ai:ananth.nandini"Muthuganesan, R."https://zbmath.org/authors/?q=ai:muthuganesan.r"Chandrasekar, V. K."https://zbmath.org/authors/?q=ai:chandrasekar.v-kSummary: The dynamical behaviour of quantum correlations captured by different forms of Measurement-Induced Nonlocality (MIN) between two atoms coupled with thermal reservoirs is investigated and compared with the entanglement. It is shown that the MIN quantities are more robust, while noise causes sudden death in entanglement. Further, we quantified the quantum correlation with weak measurement and the effect of measurement strength is observed. The role of mean photon number and weak measurements on quantum correlation is also highlighted.Utilizing adaptive boosting to detect quantum steerabilityhttps://zbmath.org/1496.810372022-11-17T18:59:28.764376Z"Song, Hong-fei"https://zbmath.org/authors/?q=ai:song.hongfei"Zhang, Jun"https://zbmath.org/authors/?q=ai:zhang.jun.5"Zhang, Hao"https://zbmath.org/authors/?q=ai:zhang.hao.4|zhang.hao.1|zhang.hao|zhang.hao.2|zhang.hao.3Summary: We use Adaptive Boosting (Adaboost) algorithm to detect the quantum steerability of the arbitrary two-qubit quantum state and predict the steerable bounds of the generalized Werner state. The results show that compared with the performance of the classifiers constructed by the support vector machine (SVM), the classifiers trained by the Adaboost are better. In particular, a high-performance classifier is obtained with partial information only measured in three fixed measurement directions. In the application of predicting the steerable bounds of the generalized Werner state, the classifiers constructed by the Adaboost predict are closer to the theoretical bounds. What is more, we give the feature selection of the high-performance classifier.Laplace transform method in one dimensional quantum mechanics on the semi infinite axishttps://zbmath.org/1496.810512022-11-17T18:59:28.764376Z"Chung, Wonsang"https://zbmath.org/authors/?q=ai:chung.won-sang"Kim, Yeounju"https://zbmath.org/authors/?q=ai:kim.yeounju"Kwon, Jeongmin"https://zbmath.org/authors/?q=ai:kwon.jeongminSummary: In this paper we discuss the Laplace transform method for solving one dimensional Schrödinger equation in a semi infinite axis. As examples we discuss the delta potential, quantum bouncer, Coulomb-like potential and half harmonic potential.A 3d gauge theory/quantum \(K\)-theory correspondencehttps://zbmath.org/1496.810852022-11-17T18:59:28.764376Z"Jockers, Hans"https://zbmath.org/authors/?q=ai:jockers.hans"Mayr, Peter"https://zbmath.org/authors/?q=ai:mayr.peterSummary: The 2d gauged linear sigma model (GLSM) gives a UV model for quantum cohomology on a Kähler manifold \(X\), which is reproduced in the IR limit. We propose and explore a 3d lift of this correspondence, where the UV model is the \(\mathcal{N} = 2\) supersymmetric 3d gauge theory and the IR limit is given by Givental's permutation equivariant quantum K-theory on \(X\). This gives a one-parameter deformation of the 2d GLSM/quantum cohomology correspondence and recovers it in a small radius limit. We study some novelties of the 3d case regarding integral BPS invariants, chiral rings, deformation spaces and mirror symmetry.Constructing the bulk at the critical point of three-dimensional large \(N\) vector theorieshttps://zbmath.org/1496.810862022-11-17T18:59:28.764376Z"Johnson, Celeste"https://zbmath.org/authors/?q=ai:johnson.celeste"Mulokwe, Mbavhalelo"https://zbmath.org/authors/?q=ai:mulokwe.mbavhalelo"Rodrigues, João P."https://zbmath.org/authors/?q=ai:rodrigues.joao-pSummary: In the context of the \(AdS_4/CFT_3\) correspondence between higher spin fields and vector theories, we use the constructive bilocal fields based approach to this correspondence, to demonstrate, at the \textit{IR} critical point of the interacting vector theory and directly in the bulk, the removal of the \(\Delta = 1\) (\(s = 0\)) state from the higher spins field spectrum, and to exhibit simple Klein-Gordon higher spin Hamiltonians. The bulk variables and higher spin fields are obtained in a simple manner from boundary bilocals, by the change of variables previously derived for the \textit{UV} critical point (in momentum space), together with a field redefinition.The \(P(\phi)_2\) Euclidean quantum field theory as classical statistical mechanics. I.https://zbmath.org/1496.820142022-11-17T18:59:28.764376Z"Guerra, F."https://zbmath.org/authors/?q=ai:guerra.francesco"Rosen, L."https://zbmath.org/authors/?q=ai:rosen.lon-m"Simon, B."https://zbmath.org/authors/?q=ai:simon.barrySee the joint review of part II [the authors, ibid. 101, No. 2, 191--259 (1975; Zbl 1495.82015)].Abstract strongly convergent variants of the proximal point algorithmhttps://zbmath.org/1496.900612022-11-17T18:59:28.764376Z"Sipoş, Andrei"https://zbmath.org/authors/?q=ai:sipos.andreiSummary: We prove an abstract form of the strong convergence of the Halpern-type and Tikhonov-type proximal point algorithms in CAT(0) spaces. In addition, we derive uniform and computable rates of metastability (in the sense of Tao) for these iterations using proof mining techniques.Characterization results of solutions in interval-valued optimization problems with mixed constraintshttps://zbmath.org/1496.900692022-11-17T18:59:28.764376Z"Treanţă, Savin"https://zbmath.org/authors/?q=ai:treanta.savinSummary: In this paper, we establish some characterization results of solutions associated with a class of interval-valued optimization problems with mixed constraints. More precisely, we investigate the connections between the LU-optimal solutions of the considered interval-valued variational control problem and the saddle-points associated with an interval-valued Lagrange functional corresponding to a modified interval-valued variational control problem. The main derived resuts are accompanied by illustrative examples.Existence of solutions of set-valued strong vector equilibrium problemshttps://zbmath.org/1496.901042022-11-17T18:59:28.764376Z"Ram, Tirth"https://zbmath.org/authors/?q=ai:ram.tirth"Lal, Parshotam"https://zbmath.org/authors/?q=ai:lal.parshotamSummary: In this paper, we considered set-valued strong vector equilibrium problems and obtained some existence results with and without compactness assumptions in Hausdorff topological vector spaces ordered by a cone. Further, we established some existence results by making use of self-segment dense set, a special type of dense set. Our results in this paper are new which can be considered as a generalization of many known results in the literature.Sub-linear convergence of a stochastic proximal iteration method in Hilbert spacehttps://zbmath.org/1496.901102022-11-17T18:59:28.764376Z"Eisenmann, Monika"https://zbmath.org/authors/?q=ai:eisenmann.monika"Stillfjord, Tony"https://zbmath.org/authors/?q=ai:stillfjord.tony"Williamson, Måns"https://zbmath.org/authors/?q=ai:williamson.mansSummary: We consider a stochastic version of the proximal point algorithm for convex optimization problems posed on a Hilbert space. A typical application of this is supervised learning. While the method is not new, it has not been extensively analyzed in this form. Indeed, most related results are confined to the finite-dimensional setting, where error bounds could depend on the dimension of the space. On the other hand, the few existing results in the infinite-dimensional setting only prove very weak types of convergence, owing to weak assumptions on the problem. In particular, there are no results that show strong convergence with a rate. In this article, we bridge these two worlds by assuming more regularity of the optimization problem, which allows us to prove convergence with an (optimal) sub-linear rate also in an infinite-dimensional setting. In particular, we assume that the objective function is the expected value of a family of convex differentiable functions. While we require that the full objective function is strongly convex, we do not assume that its constituent parts are so. Further, we require that the gradient satisfies a weak local Lipschitz continuity property, where the Lipschitz constant may grow polynomially given certain guarantees on the variance and higher moments near the minimum. We illustrate these results by discretizing a concrete infinite-dimensional classification problem with varying degrees of accuracy.