Recent zbMATH articles in MSC 46Lhttps://zbmath.org/atom/cc/46L2021-05-28T16:06:00+00:00WerkzeugCompact group actions on operator algebras and their spectra.https://zbmath.org/1459.460642021-05-28T16:06:00+00:00"Peligrad, Costel"https://zbmath.org/authors/?q=ai:peligrad.costelSummary: We consider a class of dynamical systems with compact non-abelian groups that include C*-, W*- and multiplier dynamical systems. We prove results that relate the algebraic properties such as simplicity or primeness of the fixed point algebras to the spectral properties of the action, including the Connes and strong Connes spectra.Permutations, tensor products, and Cuntz algebra automorphisms.https://zbmath.org/1459.053472021-05-28T16:06:00+00:00"Brenti, Francesco"https://zbmath.org/authors/?q=ai:brenti.francesco"Conti, Roberto"https://zbmath.org/authors/?q=ai:conti.roberto.1Summary: We study the reduced Weyl groups of the Cuntz algebras \(\mathcal{O}_n\) from a combinatorial point of view. Their elements correspond bijectively to certain permutations of \(n^r\) elements, which we call stable. We mostly focus on the case \(r = 2\) and general \(n\). A notion of rank is introduced, which is subadditive in a suitable sense. Being of rank 1 corresponds to solving an equation which is reminiscent of the Yang-Baxter equation. Symmetries of stable permutations are also investigated, along with an immersion procedure that allows to obtain stable permutations of \((n + 1)^2\) objects starting from stable permutations of \(n^2\) objects. A complete description of stable transpositions and of stable 3-cycles of rank 1 is obtained, leading to closed formulas for their number. Other enumerative results are also presented which yield lower and upper bounds for the number of stable permutations.The modular Gromov-Hausdorff propinquity.https://zbmath.org/1459.460672021-05-28T16:06:00+00:00"Latrémolière, Frédéric"https://zbmath.org/authors/?q=ai:latremoliere.fredericSummary: Motivated by the quest for an analytic framework to study classes of \(\mathrm{C}^*\)-algebras and associated structures as geometric objects, we introduce a metric on Hilbert modules equipped with a generalized form of a differential structure, thus extending Gromov-Hausdorff convergence theory to vector bundles and quantum vector bundles -- not convergence as total space but indeed as quantum vector bundle. Our metric is new even in the classical picture, and creates a framework for the study of the moduli spaces of modules over \(\mathrm{C}^*\)-algebras from a metric perspective. We apply our construction, in particular, to the continuity of Heisenberg modules over quantum \(2\)-tori.Buzano inequality in inner product \(C^*\)-modules via the operator geometric mean.https://zbmath.org/1459.460522021-05-28T16:06:00+00:00"Fujii, Jun Ichi"https://zbmath.org/authors/?q=ai:fujii.jun-ichi"Fujii, Masatoshi"https://zbmath.org/authors/?q=ai:fujii.masatoshi"Seo, Yuki"https://zbmath.org/authors/?q=ai:seo.yukiSummary: In this paper, by means of the operator geometric mean, we show a Buzano type inequality in an inner product $C^*$-module, which is an extension of the Cauchy-Schwarz inequality in an inner product $C^*$-module.\(K\)-stability of continuous \(C(X)\)-algebras.https://zbmath.org/1459.460662021-05-28T16:06:00+00:00"Seth, Apurva"https://zbmath.org/authors/?q=ai:seth.apurva"Vaidyanathan, Prahlad"https://zbmath.org/authors/?q=ai:vaidyanathan.prahladSummary: A C*-algebra is said to be \(K\)-stable if its nonstable \(K\)-groups are naturally isomorphic to the usual \(K\)-theory groups. We study continuous \(C(X)\)-algebras, each of whose fibers are \(K\)-stable. We show that such an algebra is itself \(K\)-stable under the assumption that the underlying space \(X\) is compact, metrizable, and of finite covering dimension.Squared-norm empirical processes.https://zbmath.org/1459.600542021-05-28T16:06:00+00:00"Vu, Vincent Q."https://zbmath.org/authors/?q=ai:vu.vincent-q"Lei, Jing"https://zbmath.org/authors/?q=ai:lei.jingSummary: This note extends a result of Mendelson on the supremum of a quadratic process to squared norms of functions taking values in a Banach space. Our method of proof is a reduction by a symmetrization argument and simple observation about the additivity of the generic chaining functional. We demonstrate an application to positive linear functionals of the sample covariance matrix and the apparent variance explained by principal components analysis (PCA).Exactness and SOAP of crossed products via Herz-Schur multipliers.https://zbmath.org/1459.420142021-05-28T16:06:00+00:00"McKee, Andrew"https://zbmath.org/authors/?q=ai:mckee.andrew"Turowska, Lyudmila"https://zbmath.org/authors/?q=ai:turowska.lyudmila-bSummary: Given a \(C^\ast \)-dynamical system \((A, G, \alpha)\), with \(G\) a discrete group, Schur \(A\)-multipliers and Herz-Schur \((A, G, \alpha)\)-multipliers are used to implement approximation properties, namely exactness and the strong operator approximation property (SOAP), of \(A \rtimes_{\alpha , r} G\). The resulting characterisations of exactness and SOAP of \(A \rtimes_{\alpha , r} G\) generalise the corresponding statements for the reduced group \(C^\ast \)-algebra.Cyclic-homology Chern-Weil theory for families of principal coactions.https://zbmath.org/1459.580032021-05-28T16:06:00+00:00"Hajac, Piotr M."https://zbmath.org/authors/?q=ai:hajac.piotr-m"Maszczyk, Tomasz"https://zbmath.org/authors/?q=ai:maszczyk.tomasz.1Summary: Viewing the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model, we construct the cyclic-homology Chern-Weil homomorphism. To realize the thus constructed Chern-Weil homomorphism as a Cartan model of the homomorphism tautologically induced by the classifying map on cohomology, we replace the unital subalgebra of coaction-invariants by its natural H-unital nilpotent extension (row extension). Although the row-extension algebra provides a drastically different model of the cyclic object, we prove that, for any row extension of any unital algebra over a commutative ring, the row-extension Hochschild complex and the usual Hochschild complex are chain homotopy equivalent. It is the discovery of an explicit homotopy formula that allows us to improve the homological quasi-isomorphism arguments of \textit{J.-L. Loday} [Cyclic homology. Berlin: Springer-Verlag (1992; Zbl 0780.18009)] and \textit{M. Wodzicki} [C. R. Acad. Sci., Paris, Sér. I 306, No. 9, 399--403 (1988; Zbl 0637.16014)]. We work with families of principal coactions, and instantiate our noncommutative Chern-Weil theory by computing the cotrace space and analyzing a dimension-drop-like effect in the spirit of \textit{P. Feng} and \textit{B. Tsygan} [Commun. Math. Phys. 140, No. 3, 481--521 (1991; Zbl 0743.17020)] for the quantum-deformation family of the standard quantum Hopf fibrations.Noncommutative Khintchine inequalities in interpolation spaces of \(L_p\)-spaces.https://zbmath.org/1459.460592021-05-28T16:06:00+00:00"Cadilhac, Léonard"https://zbmath.org/authors/?q=ai:cadilhac.leonardThe author proves a series of results on Khintchine inequalities in symmetric spaces obtained by interpolation between \(L^p\)-spaces. In particular, he proves that the lower Khintchine
inequality in \(L^p\) for \(p < 2\) implies the lower Khintchine inequalities for all interpolation
spaces between \(L^p\) and \(L^\infty\) with a decomposition that does not depend on the space. Using this result, it is shown that Khintchine inequalities hold in \(L^{1,\infty}\). The author also clarifies the situation with Khintchine inequalities in \(L^{2,\infty}\). Neither of the usual closed formulas for Khintchine inequalities holds in this case, but the author finds a new, deterministic equivalent of the usual \(RC\)-norm in \(L^{2,\infty}\) that works in all interpolation spaces between \(L^p\) spaces, unifying the cases \(p<2\) and \(p>2\). Applications to martingale inequalities are also given.
Reviewer: Stanisław Goldstein (Łódź)Pfaffian point processes from free fermion algebras: perfectness and conditional measures.https://zbmath.org/1459.601092021-05-28T16:06:00+00:00"Koshida, Shinji"https://zbmath.org/authors/?q=ai:koshida.shinjiSummary: The analogy between determinantal point processes (DPPs) and free fermionic calculi is well-known. We point out that, from the perspective of free fermionic algebras, Pfaffian point processes (PfPPs) naturally emerge, and show that a positive contraction acting on a ``doubled'' one-particle space with an additional structure defines a unique PfPP. Recently, Olshanski inverted the direction from free fermions to DPPs, proposed a scheme to construct a fermionic state from a quasi-invariant probability measure, and introduced the notion of perfectness of a probability measure. We propose a method to check the perfectness and show that Schur measures are perfect as long as they are quasi-invariant under the action of the symmetric group. We also study conditional measures for PfPPs associated with projection operators. Consequently, we show that the conditional measures are again PfPPs associated with projection operators onto subspaces explicitly described.\(L\)-effect algebras.https://zbmath.org/1459.080032021-05-28T16:06:00+00:00"Rump, Wolfgang"https://zbmath.org/authors/?q=ai:rump.wolfgang"Zhang, Xia"https://zbmath.org/authors/?q=ai:zhang.xiaIn this paper, \(L\)-effect algebras are introduced as a class of \(L\)-algebras which specialize to all known generalizations of effect algebras with a \(\wedge\)-semilattice structure. Moreover, \(L\)-effect algebras \(X\) arise in connection with quantum sets and Frobenius algebras. The translates of \(X\) in the self-similar closure \(S(X)\) form a covering, and the structure of \(X\) is shown to be equivalent to the compatibility of overlapping translates. A second characterization represents an \(L\)-effect algebra in the spirit of closed categories. As an application, it is proved that every lattice effect algebra is an interval in a right \(\ell\)-group, the structure group of the corresponding \(L\)-algebra. A block theory for generalized lattice effect algebras, and the existence of a generalized OML as the subalgebra of sharp elements are derived from this description.
Reviewer: Jafar Pashazadeh (Bonab)Second order deformations of group commuting squares and Hadamard matrices.https://zbmath.org/1459.460582021-05-28T16:06:00+00:00"Nicoara, Remus"https://zbmath.org/authors/?q=ai:nicoara.remus"White, Joseph"https://zbmath.org/authors/?q=ai:white.joseph-p|white.joseph-h|white.joseph-lSummary: In [Indiana Univ. Math. J. 60, No.~3, 847--857 (2011; Zbl 1253.46069)] the first author introduced second order necessary conditions for a commuting square to admit sequential deformations in the moduli space of non-isomorphic commuting squares. In this paper we investigate these conditions for commuting squares \(\mathfrak{C}_G\) constructed from finite groups \(G\). We are especially interested in the case \(G=\mathbb{Z}_n\), since deformations of \(\mathfrak{C}_{\mathbb{Z}_n}\) correspond to deformations of the Fourier matrix \(F_n\) in the moduli space of non-equivalent complex Hadamard matrices. We show that for \(G=\mathbb{Z}_n\) the second order conditions follow automatically from the first order conditions, but this is not necessarily true for other finite abelian groups \(G\). Our result gives a complete description of the second order deformations of the Fourier matrix \(F_n\) in the moduli space of non-equivalent complex Hadamard matrices.Jordan operator algebras revisited.https://zbmath.org/1459.460652021-05-28T16:06:00+00:00"Blecher, David P."https://zbmath.org/authors/?q=ai:blecher.david-p"Wang, Zhenhua"https://zbmath.org/authors/?q=ai:wang.zhenhuaSummary: Jordan operator algebras are norm-closed spaces of operators on a Hilbert space with \(a^2\in A\) for all \(a\in A\). In two recent papers by the authors and Neal, a theory for these spaces was developed. It was shown there that much of the theory of associative operator algebras, in particular, surprisingly much of the associative theory from several recent papers of the first author and coauthors, generalizes to Jordan operator algebras. In the present paper we complete this task, giving several results which generalize the associative case in these papers, relating to unitizations, real positivity, hereditary subalgebras, and a couple of other topics. We also solve one of the three open problems stated at the end of our earlier joint paper on Jordan operator algebras.Quadratic open quantum harmonic oscillator.https://zbmath.org/1459.470172021-05-28T16:06:00+00:00"Dhahri, Ameur"https://zbmath.org/authors/?q=ai:dhahri.ameur"Fagnola, Franco"https://zbmath.org/authors/?q=ai:fagnola.franco"Yoo, Hyun Jae"https://zbmath.org/authors/?q=ai:yoo.hyun-jaeThe Lie algebra \(\mathfrak{sl}_2\) has basis \(\{ B^+, B, M \}\) such that the commutation relations
\[
[ B, B^+ ] = M, \quad [ M, B ] = -2 B \quad \text{and} \quad [ M, B^+ ] = 2 B^+
\]
are satisfied. Within quantum probability, this algebra is known as the renormalised square of white noise algebra. It has a representation on the simple Fock space \(\mathsf{h} = \ell^2( \mathbb{Z}_+ )\) given by the setting
\[
B e_n = \omega_n^{1 / 2} e_{n - 1}, \quad B^+ e_n = \omega_{n + 1}^{1 / 2} e_{n + 1} \quad \text{and} \quad M e_n = ( 2 n + r ) e_n \quad \text{for all }n \geq 0,
\]
where \(\omega_n = n ( n + r - 1 )\) for some \(r > 0\) and \(( e_n )_{n \geq 0}\) is the canonical basis of \(\mathsf{h}\), and extending each operator to the natural domain of the number operator \(N\), where \(N e_n = n e_n\) for all \(n \geq 0\).
For real parameters \(\lambda\), \(\mu\), \(\zeta^+\) and \(\zeta^-\), with \(\mu \geq \lambda > 0\), the formal Gorini-Kossakowski-Sudarshan-Lindblad generator
\[
\begin{multlined}
\mathcal{L} : x \mapsto -\frac{\lambda^2}{2} ( B B^+ x - 2 B x B^+ + x B B^+ ) \\
-\frac{\mu^2}{2} ( B^+ B x - 2 B^+ x B + x B^+ B ) \\
+ \mathrm{i} [ \zeta^+ B B^+ + \zeta^- B^+ B, x ]
\end{multlined}
\]
corresponds at \(x \in B( \mathsf{h} )\) to the sesquilinear form
\[
\rlap{-}\mathcal{L}( x ) : ( u, v ) \mapsto \langle G u, x v \rangle +\lambda^2 \langle B^+ u, x B^+ v \rangle +\mu^2 \langle B u, x B v \rangle +
\langle u, x G v \rangle
\]
on the domain of \(N^2\), where the operator
\[
G = -\frac{\lambda^2}{2} B B^+ - \frac{\mu^2}{2} B^+ B -\mathrm{i} ( \zeta^+ B B^+ + \zeta^- B^+ B ).
\]
The minimal quantum Markov semigroup \(\mathcal{T}\) corresponding to this formal generator describes the dynamics of the quadratic open quantum harmonic oscillator.
The authors explain why this semigroup exists and is unique. When \(\mu > \lambda\), they give an explicit description of its unique invariant normal state \(\rho\) and show that the action of the predual semigroup \(\mathcal{T}_*\) on any initial state converges to \(\rho\). For \(\lambda = \mu\), it is shown that the semigroup is transient and so has no invariant state.
For a suitable choice of parameters, the semigroup \(\mathcal{T}\) is shown to be unitarily equivalent to the quantum Markov semigroup for a two-photon absorption and emission process studied elsewhere.
The invariant state \(\rho\) is used to compress the semigroup \(\mathcal{T}\) to obtain a contraction semigroup \(T\) on the space of Hilbert-Schmidt operators on \(\mathsf{h}\). The paper concludes with an analysis of the spectral gap for the generator of \(T\), with an explicit formula obtained for a range of parameters. In particular, if \(r > 2 \lambda^2 / ( \mu^2 - \lambda^2 )\), then the spectral gap is shown to equal
\[
\lambda^2 + \frac{r}{2} ( \mu^2 - \lambda^2 ).
\]
The paper is well written and the arguments are clear. Many of the proofs are obtained by applying results on quantum dynamical semigroups established elsewhere, together with some inspiration from the classical literature.
Reviewer: Alexander Belton (Lancaster)Entropy theory for the parametrization of the equilibrium states of Pimsner algebras.https://zbmath.org/1459.460552021-05-28T16:06:00+00:00"Kakariadis, Evgenios T. A."https://zbmath.org/authors/?q=ai:kakariadis.evgenios-t-aSummary: We consider Pimsner algebras that arise from C*-correspondences of finite rank, as dynamical systems with their rotational action. We revisit the Laca-Neshveyev classification of their equilibrium states at positive inverse temperature along with the parametrizations of the finite and the infinite parts simplices by tracial states on the diagonal. The finite rank entails an entropy theory that shapes the KMS-structure. We prove that the infimum of the tracial entropies dictates the critical inverse temperature, below which there are no equilibrium states for all Pimsner algebras. We view the latter as the entropy of the ambient C*-correspondence. This may differ from what we call strong entropy, above which there are no equilibrium states of infinite type. In particular, when the diagonal is abelian then the strong entropy is a maximum critical temperature for those. In this sense we complete the parametrization method of Laca-Raeburn and unify a number of examples in the literature.Classification of regular subalgebras of the hyperfinite II\(_1\) factor.https://zbmath.org/1459.460542021-05-28T16:06:00+00:00"Popa, Sorin"https://zbmath.org/authors/?q=ai:popa.sorin-teodor"Shlyakhtenko, Dimitri"https://zbmath.org/authors/?q=ai:shlyakhtenko.dimitri-l"Vaes, Stefaan"https://zbmath.org/authors/?q=ai:vaes.stefaanThe present articles achieves three aims. First it classifies regular subalgebras of the amenable II\(_1\) factor \(B \subset R\) that satisfy the freeness condition \(B' \cap R = \mathcal{Z}(B)\) in terms of an associated discrete measurable groupoid. Second, in order to obtain this result, it proves a cocycle vanishing theorem for free actions of amenable discrete measured groupoids on II\(_1\) von Neumann algebras. Third, to illustrate the complexity of the resulting classification, the authors obtain results on the model-theoretic complexity of classifying amenable discrete measured groupoids.
It is known by work of \textit{A. Connes} et al. [Ergodic Theory Dyn. Syst. 1, 431--450 (1981; Zbl 0491.28018)] that the amenable II\(_1\) factor \(R\) has a unique Cartan subalgebra up to isomorphism, that is, a regular von Neumann subalgebra \(A \subset R\) such that \(A' \cap R = A\). This is essentially a uniqueness theorem for the amenable ergodic II\(_1\) equivalence relation combined with a cocyle vanishing theorem. Indeed, \textit{J. Feldman} and \textit{C. C. Moore} [Trans. Am. Math. Soc. 234, 289--324 (1977; Zbl 0369.22009); ibid. 234, 325--359 (1977; Zbl 0369.22010)] proved that \(A \subset R\) is isomorphic to \(\mathrm{L}^\infty(X) \subset \mathrm{L}(\mathcal{R}, \sigma)\) for a II\(_1\) equivalence relation \(\mathcal{R}\) twisted by a 2-cocycle \(\sigma\). The starting point of the present classification result is an analogue description of regular inclusions \(B \subset R\) satisfying \(B' \cap R = \mathcal{Z}(B)\) as a twisted crossed product of an amenable discrete measured groupoid acting freely on \(B\). The cocycle vanishing result used to conclude the authors' description of such inclusions subsumes many previously known results in the framework of groups and equivalence relations, and at the same time uses those in its proof.
The article is clearly structured and well written. In particular, it addresses the problem context in a concise way.
Reviewer: Sven Raum (Stockholm)On the index of product systems of Hilbert modules.https://zbmath.org/1459.460602021-05-28T16:06:00+00:00"Kečkić, Dragoljub J."https://zbmath.org/authors/?q=ai:keckic.dragoljub-j"Vujošević, Biljana"https://zbmath.org/authors/?q=ai:vujosevic.biljanaSummary: In this note we prove that the set of all uniformly continuous units on a product system over a
$C^*$ algebra $\mathcal B$ can be endowed with the structure of left right $\mathcal B$-$\mathcal B$
Hilbert module after identifying similar units by the suitable equivalence relation. We use this construction
to define the index of the initial product system, and prove that it is the generalization of earlier defined
indices by \textit{W. Arveson} [Continuous analogues of Fock space. Providence, RI: American Mathematical Society (AMS) (1989; Zbl 0697.46035)] (in the case $\mathcal B=\mathbb C$) and \textit{M. Skeide} [Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9, No. 4, 617--655 (2006; Zbl 1119.46051)] (in the case of spatial product system).
We prove that such defined index is a covariant functor from the category of continuous product systems to
the category of $\mathcal B$ bimodules. We also prove that the index is subadditive with respect to the outer
tensor product of product systems, and prove additional properties of the index of product systems that can
be embedded into a spatial one.Commutators and images of noncommutative polynomials.https://zbmath.org/1459.160222021-05-28T16:06:00+00:00"Brešar, Matej"https://zbmath.org/authors/?q=ai:bresar.matejThe results in this paper relate to two classical questions in ring theory. One is the Kaplansky-L'vov conjecture which asks whether the image of a non-central, non-identity polynomial on \(A=M_n(F)\) must be either all of \(A\) or the commutator space, where \(F\) is an infinite field. The paper's main result in this direction is that if \(A\) is an algebra over the infinite field \(F\) and if \(A=[A,A]\) then the image of every non-constant polynomial on \(A\) is all of \(A\).
The second classical question is the Waring problem for matrices, asking whether an arbitrary matrix can be expressed as a sum of a given number of \(k\)-th powers. We quote some of the paper's results on an analogous problem. The ancillary results are also interesting, but we will not quote them.
Let \(F\) be a field of characteristic zero, \(C\) an \(F\)-algebra and \(f\) a polynomial which is neither central nor an identity for \(A=M_n(C)\). Then every commutator in \(A\) is a sum of at most 7788 elements of \(f(A)-f(A)\). If \(C\) is an integral domain the number can be reduced to 1958. If \(C\) is a field it can be reduced further to 68, and if the field is characteristic 0 and algebraically closed then the number can be reduced to 4. These numbers are not presumed to be best possible.
Reviewer: Allan Berele (Chicago)Banach-space operators acting on semicircular elements induced by orthogonal projections.https://zbmath.org/1459.460532021-05-28T16:06:00+00:00"Cho, Ilwoo"https://zbmath.org/authors/?q=ai:cho.ilwooSummary: The main purposes of this paper are (i) to construct-and-study weighted-semicircular elements from mutually orthogonal \(|\mathbb{Z}|\)-many projections, and the Banach \(*\)-probability space \(\mathbb{L}_Q\) generated by these operators, (ii) to establish \(*\)-isomorphisms on \(\mathbb{L}_Q\) induced by shifting processes on the set \(\mathbb{Z}\) of integers, (iii) to consider how the \(*\)-isomorphisms of (ii) generate Banach-space adjointable operators acting on the Banach \(*\)-algebra \(\mathbb{L}_Q\), (iv) to investigate operator-theoretic properties of the operators of (iii), and (v) to study how the Banach-space operators of (iii) distort the original free-distributional data on \(\mathbb{L}_Q\). As application, one can check how the semicircular law is distorted by our Banach-space operators on \(\mathbb{L}_Q\).A hyperfinite factor which is not an injective \(\mathrm{C}^\ast\)-algebra.https://zbmath.org/1459.460572021-05-28T16:06:00+00:00"Wright, J. D. Maitland"https://zbmath.org/authors/?q=ai:wright.john-david-maitland"Saitô, Kazuyuki"https://zbmath.org/authors/?q=ai:saito.kazuyukiSummary: We exhibit a wild monotone complete \(\mathrm{C}^\ast\)-algebra which is a hyperfinite factor but is not an injective \(\mathrm{C}^\ast\)-algebra.Individual ergodic theorems in noncommutative Orlicz spaces.https://zbmath.org/1459.470062021-05-28T16:06:00+00:00"Chilin, Vladimir"https://zbmath.org/authors/?q=ai:chilin.vladimir-ivanovich"Litvinov, Semyon"https://zbmath.org/authors/?q=ai:litvinov.semyon-nLet \({\mathcal M}\) be a semifinite von Neumann algebra with a faithful normal semifinite trace \(\tau\), \(L^0=L^0({\mathcal M},\tau)\) the \(*\)-algebra of \(\tau\)-measurable operators affiliated with \({\mathcal M}\), and let \(L^p=L^p({\mathcal M},\tau)\), \(1\leq p\leq \infty\), be the noncommutative \(L^p\) space associated with \(({\mathcal M},\tau)\). A~linear map \(T:L^1+L^\infty \to L^1+ L^\infty\) such that \(\| T(x)\|_\infty \leq \| x\|_\infty\) for all \(x\in {\mathcal M}\) and \(\| T(x)\|_1 \leq \| x\|_1\) for all \(x\in L^1\) is called a Dunford-Schwartz operator.
Let \(\Phi\) be an Orlicz function and let \(L^\Phi =L^\Phi({\mathcal M},\tau)\) be the noncommutative Orlicz space associated with \(({\mathcal M},\tau)\). The Orlicz function satisfies a \(\Delta_2\)-condition (resp., \(\delta_2\)-condition) if there exists \(k>0\) and \(u_0\geq 0\) such that \(\Phi(2u)\leq k\Phi(u)\) for all \(u\geq u_0\), (resp., \(\Phi(2u)\leq k\Phi(u)\) for all \(u\in (0,u_0]\)).
The main result of the paper shows that, if \(T\) is a positive Dunford-Schwartz operator and \(\Phi\) is an Orlicz function satisfying the \((\delta_2, \Delta_2)\) condition, then, for all \(x\in L^\Phi\), the averages \(\frac{1}{n}\sum_{k=1}^n T^k(x)\) converge in the b.a.u.\ (bilateral almost uniform) topology to some \(\hat{x}\) in \(L^\Phi\). A~sequence \((x_n)\subset L^0\) converges in the b.a.u.\ topology to \(x\) if, for all \(\varepsilon>0\), there exists a projection \(e\in{\mathcal M}\) such that \(\tau(e^\perp)<\varepsilon\) and \(\| e(x-x_n)e\|_\infty\to 0\).
Reviewer: Franco Fagnola (Milano)The Leavitt path algebras of ultragraphs.https://zbmath.org/1459.160322021-05-28T16:06:00+00:00"Imanfar, Mostafa"https://zbmath.org/authors/?q=ai:imanfar.mostafa"Pourabbas, Abdolrasoul"https://zbmath.org/authors/?q=ai:pourabbas.abdolrasoul"Larki, Hossein"https://zbmath.org/authors/?q=ai:larki.hosseinThe purpose of this paper is to define the algebraic versions of ultragraphs \(C^*\)-algebras and Exel-Laca algebras. For an ultragraph \(\mathcal{G}=(G^0,\mathcal{G}^1,r,s)\) and unital commutative ring \(R\), a Leavitt \(G\)-family in an \(R\)-algebra \(A\) is a set \(\{p_S, s_e, s_{e^*}: S \in \mathcal{G}^0, e\in \mathcal{G}^1\}\subset A\) subject to conditions which resemble those of a Cuntz-Krieger family. The set \(\mathcal{G}^0\) is defined suitably (but not in the paper under review). The path algebra \(L_R(G)\) is defined as the \(R\)-algebra generated by a universal Leavitt \(\mathcal{G}\)-family. Within this theory one can define hereditary and saturated subsets and admissible pairs \((H,S)\) in a way that generalize the corresponding notions for classical graphs. Also the construction of the quotient ultragraph \(\mathcal{G}/(H,S)\) is used in the paper. Given an admissible pair \((H, S)\) in \(\mathcal{G}\), the authors define the Leavitt path algebra \(L_R(\mathcal{G}/(H, S))\) associated to the quotient ultragraph \(\mathcal{G}/(H, S)\) and prove two kinds of uniqueness theorems, namely the Cuntz-Krieger and the graded-uniqueness theorems, for \(L_R(\mathcal{G}/(H, S))\).
Then these uniqueness theorems are applied to analyze the ideal structure of \(L_R(G)\). It is proved that (1) every Leavitt path algebra of a directed graph can be embedded as an subalgebra in a unital ultragraph Leavitt path algebra; (2) the ultragraph Leavitt path algebra \(L_{\mathbb{C}}(\mathcal{G})\) is isomorphic to a dense \(*\)-subalgebra of \(C^*(\mathcal{G})\); (3) an ultragrpah \(\mathcal{G}\) satisfies condition (K) if and only if every basic ideal in \(L_R(\mathcal{G})\) is graded. Then this notion is used to introduce and study the algebraic analogue of Exel-Laca algebras. There are some examples proving that there are ultragraphs \(\mathcal{G}\) such that \(L_R(\mathcal{G})\) is not a (classical) Leavitt path algebra and is not either an Excel-Laca \(R\)-algebra.
Reviewer: Candido Martín González (Málaga)The index of product systems of Hilbert modules: two equivalent definitions.https://zbmath.org/1459.460612021-05-28T16:06:00+00:00"Vujošević, Biljana"https://zbmath.org/authors/?q=ai:vujosevic.biljanaSummary: We prove that a conditionally completely positive definite kernel, as the generator of completely positive definite (CPD) semigroup associated with a continuous set of units for a product system over a $C^*$-algebra $\mathcal{B}$, allows a construction of a Hilbert $\mathcal{B}$-$\mathcal{B}$ module. That construction is used to define the index of the initial product system. It is proved that such definition is equivalent to the one previously given by \textit{D.~J. Kečkić} and \textit{B.~Vujošević} [Filomat 29, No.~5, 1093--1111 (2015; Zbl 1459.46060)]. Also, it is pointed out that the new definition of the index corresponds to the one given earlier by \textit{W. Arveson} (in the case $\mathcal{B}=\mathbb{C}$) [Noncommutative dynamics and $E$-semigroups. New York, NY: Springer (2003; Zbl 1032.46001)].On a Poisson summation formula for noncommutative tori.https://zbmath.org/1459.111272021-05-28T16:06:00+00:00"Nikolaev, Igor"https://zbmath.org/authors/?q=ai:nikolaev.igor-g|nikolaev.igor-vasilievichSummary: It is proved that a maximal abelian subalgebra of the noncommutative torus commutes with the Laplace operator on a complex torus. As a corollary, one gets an analog of the Poisson summation formula for noncommutative tori.Uniformity in \(C^*\)-algebras.https://zbmath.org/1459.460512021-05-28T16:06:00+00:00"Wegert, Adam"https://zbmath.org/authors/?q=ai:wegert.adamSummary: We introduce a notion of uniform structure on the set of all representations of a given separable, not necessarily commutative \(C^*\)-algebra \(\mathfrak A\) by introducing a suitable family of metrics on the set of representations of \(\mathfrak A\) and investigate its properties. We define the noncommutative analogue of the notion of the modulus of continuity of an element in a \(C^*\)-algebra and we establish its basic properties. We also deal with morphisms of \(C^*\)-algebras by defining two notions of uniform continuity and show their equivalence.On \({}^*\)-similarity in \(C^*\)-algebras.https://zbmath.org/1459.460502021-05-28T16:06:00+00:00"Marcoux, L. W."https://zbmath.org/authors/?q=ai:marcoux.laurent-w"Radjavi, H."https://zbmath.org/authors/?q=ai:radjavi.heydar"Yahaghi, B. R."https://zbmath.org/authors/?q=ai:yahaghi.bamdad-rSummary: Two subsets \(\mathcal X\) and \(\mathcal Y\) of a unital \(C^*\)-algebra \(\mathcal A\) are said to be \({}^*\)-similar via \(s \in \mathcal A^{-1}\) if \(\mathcal Y = s^{-1} \mathcal X s\) and \(\mathcal Y^* = s^{-1} \mathcal X^* s\). We show that this relation imposes a certain structure on the sets \(\mathcal X\) and \(\mathcal Y\), and that under certain natural conditions (for example, if \(\mathcal X\) is bounded), \({}^*\)-similar sets must be unitarily equivalent. As a consequence of our main results, we present a generalized version of a well-known theorem of \textit{W. Specht} [Jahresber. Dtsch. Math.-Ver. 50, 19--23 (1940; Zbl 0023.00203)].Higher variations for free Lévy processes.https://zbmath.org/1459.460622021-05-28T16:06:00+00:00"Anshelevich, Michael"https://zbmath.org/authors/?q=ai:anshelevich.michael"Wang, Zhichao"https://zbmath.org/authors/?q=ai:wang.zhichaoSummary: For a general free Lévy process, we prove the existence of its higher variation processes as limits in distribution, and identify the limits in terms of the Lévy-Itô representation of the original process. For a general free compound Poisson process, this convergence holds almost uniformly. This implies joint convergence in distribution to a \(k\)-tuple of higher variation processes, and so the existence of \(k\)-fold stochastic integrals as almost uniform limits. If the existence of moments of all orders is assumed, the result holds for free additive (not necessarily stationary) processes and more general approximants. In the appendix we note relevant properties of symmetric polynomials in non-commuting variables.The covariant Gromov-Hausdorff propinquity.https://zbmath.org/1459.460682021-05-28T16:06:00+00:00"Latrémolière, Frédéric"https://zbmath.org/authors/?q=ai:latremoliere.fredericSummary: We extend the Gromov-Hausdorff propinquity to a metric on Lipschitz dynamical systems which are given by strongly continuous actions of proper monoids on quantum compact metric spaces via Lipschitz morphisms. We prove that the resulting distance between two Lipschitz dynamical systems is zero if and only if there exists an equivariant full quantum isometry between them. We apply our results to convergence of the dual actions on fuzzy tori to the dual actions on quantum tori. Our framework is general enough to also allow for the study of the convergence of continuous semigroups of positive linear maps and other actions of proper monoids.On the numerical index with respect to an operator.https://zbmath.org/1459.460132021-05-28T16:06:00+00:00"Kadets, Vladimir"https://zbmath.org/authors/?q=ai:kadets.vladimir-m"Martín, Miguel"https://zbmath.org/authors/?q=ai:martin-suarez.miguel"Merí, Javier"https://zbmath.org/authors/?q=ai:meri.javier"Pérez, Antonio"https://zbmath.org/authors/?q=ai:perez-hernandez.antonio"Quero, Alicia"https://zbmath.org/authors/?q=ai:quero.aliciaSummary: The aim of this paper is to study the numerical index with respect to an operator between Banach spaces. Given Banach spaces \(X\) and \(Y\), and a norm-one operator \(G\in \mathcal{L}(X,Y)\) (the space of all bounded linear operators from \(X\) to \(Y)\), the numerical index with respect to \(G$, $n_G(X,Y)\), is the greatest constant \(k\geq 0\) such that
\[
k\|T\|\leq \inf_{\delta > 0} \sup\{|y^\ast(Tx)|\colon y^\ast\in Y^\ast,\,x\in X,\,\|y^\ast\|=\|x\|=1,\,\operatorname{Re} y^\ast(Gx) > 1-\delta\}
\]
for every \(T\in \mathcal{L}(X,Y)\). Equivalently, \(n_G(X,Y)\) is the greatest constant \(k\geq 0\) such that \[\max_{|w|=1}\|G+wT\|\geq 1 + k \|T\|\] for all \(T\in \mathcal{L}(X,Y)\). Here, we first provide some tools to study the numerical index with respect to \(G\). Next, we present some results on the set \(\mathcal{N}(\mathcal{L}(X,Y))\) of the values of the numerical indices with respect to all norm-one operators in \(\mathcal{L}(X,Y)\). For instance, \( \mathcal{N}(\mathcal{L}(X,Y))=\{0\}\) when \(X\) or \(Y\) is a real Hilbert space of dimension greater than 1 and also when \(X\) or \(Y\) is the space of bounded or compact operators on an infinite-dimensional real Hilbert space. In the real case, \[\mathcal{N}(\mathcal{L}(X,\ell_p))\subseteq [0,M_p] \quad \text{and} \quad \mathcal{N}(\mathcal{L}(\ell_p,Y))\subseteq [0,M_p]\] for \(1 < p < \infty\) and for all real Banach spaces \(X\) and \(Y\), where \(M_p=\sup_{t\in[0,1]}\frac{|t^{p-1}-t|}{1+t^p}\). For complex Hilbert spaces \(H_1, H_2\) of dimension greater than 1, \(\mathcal{N}(\mathcal{L}(H_1,H_2))\subseteq \{0,1/2\}\) and the value \(1/2\) is taken if and only if \(H_1\) and \(H_2\) are isometrically isomorphic. Moreover, \( \mathcal{N}(\mathcal{L}(X,H))\subseteq [0,1/2]\) and \(\mathcal{N}(\mathcal{L}(H,Y))\subseteq [0,1/2]\) when \(H\) is a complex infinite-dimensional Hilbert space and \(X\) and \(Y\) are arbitrary complex Banach spaces. Also, \( \mathcal{N}(\mathcal{L}(L_1(\mu_1),L_1(\mu_2)))\subseteq \{0,1\}\) and \(\mathcal{N}(\mathcal{L}(L_\infty(\mu_1),L_\infty(\mu_2)))\subseteq \{0,1\}\) for arbitrary \(\sigma\)-finite measures \(\mu_1\) and \(\mu_2\), in both the real and the complex cases. Also, we show that the Lipschitz numerical range of Lipschitz maps from a Banach space to itself can be viewed as the numerical range of convenient bounded linear operators with respect to a bounded linear operator. Further, we provide some results which show the behaviour of the value of the numerical index when we apply some Banach space operations, such as constructing diagonal operators between \(c_0$-, $\ell_1\)-, or \(\ell_\infty\)-sums of Banach spaces, composition operators on some vector-valued function spaces, taking the adjoint to an operator, and composition of operators.The covariant Stone-von Neumann theorem for actions of abelian groups on \(C^* \)-algebras of compact operators.https://zbmath.org/1459.460632021-05-28T16:06:00+00:00"Huang, Leonard"https://zbmath.org/authors/?q=ai:huang.leonard-t"Ismert, Lara"https://zbmath.org/authors/?q=ai:ismert.laraSummary: In this paper, we formulate and prove a version of the Stone-von Neumann Theorem for every \(C^* \)-dynamical system of the form \(\left( G,{\mathbb{K}} \left( {\mathcal{H}} \right) ,\alpha \right) \), where \(G\) is a locally compact Hausdorff abelian group and \({\mathcal{H}}\) is a Hilbert space. The novelty of our work stems from our representation of the Weyl Commutation Relation on Hilbert \({\mathbb{K}} \left( {\mathcal{H}} \right) \)-modules, instead of just Hilbert spaces, and our introduction of two additional commutation relations, which are necessary to obtain a uniqueness theorem. Along the way, we apply one of our basic results on Hilbert \(C^* \)-modules to significantly shorten the length of \textit{I. Raeburn}'s well-known proof of Takai-Takesaki Duality [Proc. Edinb. Math. Soc., II. Ser. 31, No. 2, 321--330 (1988; Zbl 0674.46038)].Properties of states on Weyl algebra with variable multiplication law.https://zbmath.org/1459.810612021-05-28T16:06:00+00:00"Ługiewicz, Piotr"https://zbmath.org/authors/?q=ai:lugiewicz.piotr"Jakóbczyk, Lech"https://zbmath.org/authors/?q=ai:jakobczyk.lech"Frydryszak, Andrzej"https://zbmath.org/authors/?q=ai:frydryszak.andrzej-mSummary: We consider possible quantum effects for infinite systems implied by variations of the multiplication law in the algebra of observables. Using the algebraic formulation of quantum theory, we study the behavior of states \(\omega\) under changes in the defining relations of the canonical commutation relations (CCR-algebra). These defining relations of the multiplication law depend explicitly on the symplectic form \(\sigma \), which encodes commutation relations of canonical field operators. We consider the change in this form given by simple rescaling of \(\sigma\) by a positive parameter \(h\). We analyze to what extent changes in \(h\) preserve the original state space (this gives restrictions on the admissible changes in the scaling parameter \(h)\) and which properties have original quantum states \(\omega\) as states on the new algebra. We answer such questions for the quasi-free states. We show that any universally invariant state can be interpreted as a convex combination of Fock states with different values of constant \(h\). The second important class of states we study is the KMS-state; here, the rescaling alters in a nontrivial way the relevant dynamics. We also show that it is possible to go beyond the limits restricting the changes in \(h\), but then one has to restrict the CCR-algebra to a subalgebra.
{\copyright 2021 American Institute of Physics}Cartan subalgebras for non-principal twisted groupoid \(C^\ast\)-algebras.https://zbmath.org/1459.460492021-05-28T16:06:00+00:00"Duwenig, A."https://zbmath.org/authors/?q=ai:duwenig.anna"Gillaspy, E."https://zbmath.org/authors/?q=ai:gillaspy.elizabeth"Norton, R."https://zbmath.org/authors/?q=ai:norton.robert-m|norton.richard-a|norton.rachael-m|norton.r-l|norton.ronald-n|norton.richard-e.1"Reznikoff, S."https://zbmath.org/authors/?q=ai:reznikoff.sarah-a"Wright, S."https://zbmath.org/authors/?q=ai:wright.scott|wright.steve|wright.sarah-e|wright.stewart-v|wright.stephen-j.1|wright.sally-e|wright.samuel|wright.steven-j|wright.stephen-j|wright.sewall|wright.s-courtenay|wright.stephen-e|wright.s-i|wright.s-aSummary: Renault proved in [\textit{J. Renault}, Ir. Math. Soc. Bull. 61, 29--63 (2008; Zbl 1175.46050); Theorem 5.2] that if \(\mathcal{G}\) is a topologically principal groupoid, then \(C_0(\mathcal{G}^{(0)})\) is a Cartan subalgebra in \(C_r^\ast(\mathcal{G}, \Sigma)\) for any twist \(\Sigma\) over \(\mathcal{G}\). However, there are many groupoids which are not topologically principal, yet their (twisted) \(C^\ast\)-algebras admit Cartan subalgebras. This paper gives a dynamical description of a class of such Cartan subalgebras, by identifying conditions on a 2-cocycle \(c\) on \(\mathcal{G}\) and a subgroupoid \(\mathcal{S} \subseteq \mathcal{G}\) under which \(C_r^\ast(\mathcal{S}, c)\) is Cartan in \(C_r^\ast(\mathcal{G}, c)\). When \(\mathcal{G}\) is a discrete group, we also describe the Weyl groupoid and twist associated to these Cartan pairs, under mild additional hypotheses.Oscillation theory for the density of states of high dimensional random operators.https://zbmath.org/1459.810392021-05-28T16:06:00+00:00"Großmann, Julian"https://zbmath.org/authors/?q=ai:grossmann.julian-p"Schulz-Baldes, Hermann"https://zbmath.org/authors/?q=ai:schulz-baldes.hermann"Villegas-Blas, Carlos"https://zbmath.org/authors/?q=ai:villegas-blas.carlosSummary: Sturm-Liouville oscillation theory is studied for Jacobi operators with block entries given by covariant operators on an infinite dimensional Hilbert space. It is shown that the integrated density of states of the Jacobi operator is approximated by the winding of the Prüfer phase w.r.t. the trace per unit volume. This rotation number can be interpreted as a spectral flow in a von Neumann algebra with finite trace.On free regular and Bondesson convolution semigroups.https://zbmath.org/1459.600992021-05-28T16:06:00+00:00"Kuznetsov, A."https://zbmath.org/authors/?q=ai:kuznetsov.alexander-g|kuznetsov.andrei-nikolavich|kuznetsov.a-yu|kuznetsov.a-e|kuznetsov.alexey|kuznetsov.a-o|kuznetsov.alex-v|kuznetsov.alexander-i|kuznetsov.a-f|kuznetsov.alexander-m|kuznetsov.artem|kuznetsov.anatolij-i|kuznetsov.a-d|kuznetsov.a-s|kuznetsov.andrey-v|kuznetsov.aleksey-mikhailovich|kuznetsov.aleksandr-petrovich|kuznetsov.alexey-s|kuznetsov.a-t|kuznetsov.arseniy-i|kuznetsov.a-b|kuznetsov.aleksandr-v|kuznetsov.alexander-aSummary: Free regular convolution semigroups describe the distribution of free subordinators, while Bondesson class convolution semigroups correspond to classical subordinators with completely monotone Lévy density. We show that these two classes of convolution semigroups are in bijection with the class of complete Bernstein functions, and we establish an integral identity linking the two semigroups. We provide several explicit examples that illustrate this result.Hermitian and algebraic \(^*\)-algebras, representable extensions of positive functionals.https://zbmath.org/1459.460562021-05-28T16:06:00+00:00"Szűcs, Zsolt"https://zbmath.org/authors/?q=ai:szucs.zsolt"Takács, Balázs"https://zbmath.org/authors/?q=ai:takacs.balazsThe purpose of the paper is to investigate the connections between the structure of a (complex) *-algebra and the property that a representable positive functional has an extension from a *-subalgebra to the whole algebra. The main result is that E\(^*\)-algebras (those *-algebras for which every representable positive functional defined on an arbitrary *-subalgebra has a representable positive extension to the whole algebra) is exactly the class of Hermitian algebraic *-algebras, that is, Hermitian *-algebras in which every element generates a finite-dimensional subalgebra. Other related results are obtained.
Reviewer: Cătălin Badea (Villeneuve d'Ascq)