Recent zbMATH articles in MSC 46Lhttps://zbmath.org/atom/cc/46L2024-02-28T19:32:02.718555ZWerkzeugOn 2nd-stage quantization of quantum cluster algebrashttps://zbmath.org/1527.130222024-02-28T19:32:02.718555Z"Li, Fang"https://zbmath.org/authors/?q=ai:li.fang"Pan, Jie"https://zbmath.org/authors/?q=ai:pan.jieSummary: Motivated by the phenomenon that compatible Poisson structures on a cluster algebra play a key role on its quantization (that is, quantum cluster algebra), we introduce the 2nd-stage quantization of a quantum cluster algebra, which means the correspondence between compatible Poisson structures of the quantum cluster algebra and its 2nd-stage quantized cluster algebras. Based on this observation, we find that a quantum cluster algebra possesses a mutually alternating quantum cluster algebra such that their 2nd-stage quantization can be essentially the same.
As an example, we give the 2nd-stage quantized cluster algebra \(A_{p,q}(SL(2))\) of \(Fun_{\mathbb{C}}(SL_q(2))\) in {\S}7.1 and show that it is a non-trivial 2nd-stage quantization, which may be realized as a parallel supplement to two parameters quantization of the general quantum group. As another example, we present a class of quantum cluster algebras with coefficients which possess a non-trivial 2nd-stage quantization. In particular we obtain a class of quantum cluster algebras from surfaces with coefficients which possess non-trivial 2nd-stage quantization. Finally, we prove that the compatible Poisson structure of a quantum cluster algebra without coefficients is always a locally standard Poisson structure. Following this, it is shown that the 2nd-stage quantization of a quantum cluster algebra without coefficients is in fact trivial.Reverse Cholesky factorization and tensor products of nest algebrashttps://zbmath.org/1527.150092024-02-28T19:32:02.718555Z"Paulsen, Vern I."https://zbmath.org/authors/?q=ai:paulsen.vern-ival"Woerdeman, Hugo J."https://zbmath.org/authors/?q=ai:woerdeman.hugo-jSummary: We prove that every positive semidefinite matrix over the natural numbers that is eventually 0 in each row and column can be factored as the product of an upper triangular matrix times a lower triangular matrix. We also extend some known results about factorization with respect to tensor products of nest algebras. Our proofs use the theory of reproducing kernel Hilbert spaces.Almost flat relative vector bundles and the almost monodromy correspondencehttps://zbmath.org/1527.190022024-02-28T19:32:02.718555Z"Kubota, Yosuke"https://zbmath.org/authors/?q=ai:kubota.yosukeSummary: In this paper, we introduce the notion of almost flatness for (stably) relative bundles on a pair of topological spaces and investigate basic properties of it. First, we show that almost flatness of topological and smooth sense are equivalent. This provides a construction of an almost flat stably relative bundle on enlargeable manifolds. Second, we show the almost monodromy correspondence, that is, a correspondence between almost flat (stably) relative bundles and (stably) relative quasi-representations of the fundamental group.Proper proximality in non-positive curvaturehttps://zbmath.org/1527.200662024-02-28T19:32:02.718555Z"Horbez, Camille"https://zbmath.org/authors/?q=ai:horbez.camille"Huang, Jingyin"https://zbmath.org/authors/?q=ai:huang.jingyin"Lécureux, Jean"https://zbmath.org/authors/?q=ai:lecureux.jeanSummary: Proper proximality of a countable group is a notion that was introduced by \textit{R. Boutonnet} et al. [Ann. Sci. Éc. Norm. Supér. (4) 54, No. 2, 445--482 (2021; Zbl 07360850)] as a tool to study rigidity properties of certain von Neumann algebras associated to groups or ergodic group actions. In the present paper, we establish the proper proximality of many groups acting on nonpositively curved spaces.
First, these include many countable groups \(G\) acting properly nonelementarily by isometries on a proper \(\mathrm{CAT}(0)\) space \(X\). More precisely, proper proximality holds in the presence of rank one isometries or when \(X\) is a locally thick affine building with a minimal \(G\)-action. As a consequence of Rank Rigidity, we derive the proper proximality of all countable nonelementary \(\mathrm{CAT}(0)\) cubical groups, and of all countable groups acting properly cocompactly nonelementarily by isometries on either a Hadamard manifold with no Euclidean factor, or on a 2-dimensional piecewise Euclidean \(\mathrm{CAT}(0)\) simplicial complex.
Second, we establish the proper proximality of many hierarchically hyperbolic groups. These include the mapping class groups of connected orientable finite-type boundaryless surfaces (apart from a few low-complexity cases), thus answering a question raised by Boutonnet, Ioana, and Peterson [loc. cit.]. We also prove the proper proximality of all subgroups acting nonelementarily on the curve graph.
In view of work of Boutonnet, Ioana and Peterson [loc. cit.], our results have applications to structural and rigidity results for von Neumann algebras associated to all the above groups and their ergodic actions.A type I conjecture and boundary representations of hyperbolic groupshttps://zbmath.org/1527.220092024-02-28T19:32:02.718555Z"Caprace, Pierre-Emmanuel"https://zbmath.org/authors/?q=ai:caprace.pierre-emmanuel"Kalantar, Mehrdad"https://zbmath.org/authors/?q=ai:kalantar.mehrdad"Monod, Nicolas"https://zbmath.org/authors/?q=ai:monod.nicolasThe main result of this paper is the beautiful Theorem A: Let \(G\) be a hyperbolic locally compact group admitting a uniform lattice. If \(G\) is of type I, then \(G\) has a cocompact amenable subgroup.
Thanks to Theorem D of the first author et al. [J. Eur. Math. Soc. (JEMS) 17, No. 11, 2903--2947 (2015; Zbl 1330.43002)], the above result leads to a rather precise description of \(G\): see Corollary C. Specializing Theorem B and Corollary C, the authors obtain Corollary D: Let \(\mathcal{T}\) be a locally finite tree and \(G \leq \Aut(\mathcal{T})\) be a closed nonamenable subgroup acting minimally on \(\mathcal{T}\). If \(G\) is type \(\mathrm{I}\), then the \(G\)-action on the set of ends \(\partial \mathcal{T}\) is 2-transitive.
Another important result is Theorem E: Any two boundary representations of a nonamenable hyperbolic locally compact group are weakly equivalent. This answers a question of \textit{C. Houdayer} and \textit{S. Raum} [Comment. Math. Helv. 94, No. 1, 185--219 (2019; Zbl 1468.22016)].
The paper under review contains several other interesting results that are too technical to be reported here.
Reviewer: Egle Bettio (Venezia)The bicategory of topological correspondenceshttps://zbmath.org/1527.220112024-02-28T19:32:02.718555Z"Holkar, Rohit Dilip"https://zbmath.org/authors/?q=ai:holkar.rohit-dilipSummary: It is known that a topological correspondence \((X, \lambda)\) from a locally compact groupoid with a Haar system \((G, \alpha)\) to another one, \((H, \beta)\), produces a \(\mathrm{C}^*\)-correspondence \(\mathcal{H}(X, \lambda)\) from \(\mathrm{C}^*(G, \alpha)\) to \(\mathrm{C}^*(H, \beta)\). We described the composition of two topological correspondences in one of our earlier articles. In the present article, we prove that second countable locally compact Hausdorff groupoids with Haar systems form a bicategory \(\mathfrak{T}\) when equipped with topological correspondences as 1-arrows and isomorphisms of topological correspondences as 2-arrows.
On the other hand, it well-known that \(\mathrm{C}^*\)-algebras form a bicategory \(\mathfrak{C}\) with \(\mathrm{C}^*\)v-correspondences as 1-arrows, and the unitary isomorphisms of Hilbert \(\mathrm{C}^*\)-modules that intertwine the representations serve as the 2-arrows. In this article, we show that a topological correspondence going to a \(\mathrm{C}^*\)-one is a bifunctor \(\mathfrak{T} \longrightarrow \mathfrak{C}\). Finally, we show that in the sub-bicategory of \(\mathfrak{T}\) consisting of the Macho-Stadler-O'uchi correspondences, invertible 1-arrows are exactly the groupoid equivalences.Analysis and quantum groupshttps://zbmath.org/1527.460012024-02-28T19:32:02.718555Z"Tuset, Lars"https://zbmath.org/authors/?q=ai:tuset.larsThis is a hugely ambitious book, which attempts to take the reader from the basics of functional analysis to understanding locally compact quantum groups, their (co)actions, and various technical constructions associated with them. The book comes in at over 600 pages, containing an appendix, short bibliography, and generous index.
Here a quantum group is an algebra (a \(C^*\)-algebra or von Neumann algebra) together with a coproduct, and ``extra structure'' somehow corresponding to having a group, and not a semigroup. We are interested in quantum groups which correspond, classically, to locally compact groups: so possibly infinite, and with a reasonably manageable topology. Already in this setting, the algebra and topology interact in a non-trivial way. For example, let \(G\) be locally compact group, and consider the \(C^*\)-algebra \(C_0(G)\) of continuous functions vanishing at infinity. Dualising the group product, we obtain a coproduct given by \(\Delta(f)(s,t) = f(st)\) for \(f\in C_0(G)\), where the codomain of \(\Delta\) is \(C^b(G\times G)\), which is not \(C_0(G)\otimes C_0(G) \cong C_0(G\times G)\), but rather its multiplier algebra. For von Neumann algebras, things are a little smoother as we work with \(L^\infty(G)\), and here the von Neumann tensor product gives the natural codomain for \(\Delta\) as \(L^\infty(G) \bar\otimes L^\infty(G) \cong L^\infty(G\times G)\).
For locally compact abelian groups, the classical Pontryagin duality theory defines a dual group and a generalisation of the Fourier transform. Within this quantum group formalism, the dual group to \(L^\infty(G)\) is \(VN(G)\), the group von Neumann algebra, which has a natural cocommutative coproduct. Historically, trying to realise this duality in a broader framework motivated much work, culminating in the theory of Kac algebras, see [\textit{M.~Enock} and \textit{J.-M. Schwartz}, Kac algebras and duality of locally compact groups. Preface by Alain Connes, postface by Adrian Ocneanu. Berlin: Springer-Verlag (1992; Zbl 0805.22003)]. This is a von Neumann algebra framework, where the ``extra structure'' alluded to is somewhat complicated.
We also have the notion of a compact quantum group, see [\textit{S. L. Woronowicz}, in: Quantum symmetries/ Symétries quantiques. Proceedings of the Les Houches summer school, Session LXIV, Les Houches, France, August 1 - September 8, 1995. Amsterdam: North-Holland. 845--884 (1998; Zbl 0997.46045)]. Here one starts with a unital \(C^*\)-algebra \(A\) and a coassociative coproduct \(\Delta:A\rightarrow A\otimes A\) which merely satisfies certain natural density (or ``cancellation'') conditions. Then, as theorems, one can prove the existence of an invariant Haar state and a dense Hopf \(*\)-algebra, together with a rich (unitary) corepresentation theory. There are by now some textbooks in this area, for example, [\textit{T.~Timmermann}, An invitation to quantum groups and duality. From Hopf algebras to multiplicative unitaries and beyond. Zürich: European Mathematical Society (2008; Zbl 1162.46001)] and [\textit{S.~Neshveyev} and \textit{L.~Tuset}, Compact quantum groups and their representation categories. Paris: Société Mathématique de France (SMF) (2013; Zbl 1316.46003)].
A modern framework which encompasses all these examples is that of the theory of locally compact quantum groups [\textit{J.~Kustermans} and \textit{S.~Vaes}, Ann. Sci. Éc. Norm. Supér. (4) 33, No.~6, 837--934 (2000; Zbl 1034.46508)]. For a less technical overview, see the papers [\textit{A.~Van Daele}, Proc. Natl. Acad. Sci. USA 97, No.~2, 541--546 (2000; Zbl 0984.16038)] and [\textit{J.~Kustermans} and \textit{S.~Vaes}, Proc. Natl. Acad. Sci. USA 97, No.~2, 547--552 (2000; Zbl 0978.46044)]. This is a \(C^*\)-algebraic theory where the key ``extra structure'' is to assume the existence of left and right invariant weights: from this follows the existence of a densely defined antipode, a duality theory, and much else. There is a parallel (and interacting) von Neumann algebra theory.
So the framework here is that of operator algebras, along with tensor products, multiplier algebras and so forth, and weight theory and Tomita-Takesaki theory, and unbounded operators, along perhaps with some classical measure theory as motivation. With this background, one may then introduce the definition of a locally compact quantum group, and prove fundamental properties of such objects, along with the duality theory. Then one might turn to various constructions with these new objects.
The book under review attempts all of this! The first 11~chapters give a fast-paced, but self-contained, account of functional analysis (Banach spaces, and bases), operators on Hilbert spaces, some \(C^*\)-algebra theory (the GNS construction and so forth). This is followed by some von Neumann algebra theory, including type classification, and then an account of tensor products, and unbounded operators. Then Tomita-Takesaki theory is treated, followed by the classification of type~III factors. While the clear end-goal is to develop the theory required for the quantum group applications, the inclusion of the chapter on bases in Banach spaces, for example, hints at the ambition to do more than simply to only collect needed background, or to merely give an ``introduction''. These initial chapters do, in some sense, form an independent whole.
Chapter 12 is devoted to the definition, and core properties, of locally compact quantum groups. This material is presented at a fast pace: everything is proved, but in a terse way, and perhaps without much motivation. It is nice to see this material collected (perhaps for the first time) in book form. Chapter~13 explores the commutative and cocommutative cases, along with maximal \(C^*\)-completions (often termed universal quantum groups) and some aspects of amenability.
Attention then turns to various constructions. First is the crossed product construction, which is explored in the classical context in Chapter~14, that is, the continuous action of a group on a \(C^*\)-algebra. Takesaki-Takai duality (that is, a duality theory for abelian groups) and Landstad theory (how to recognise an algebra which is isomorphic to a crossed-product) are treated. Chapter~15 looks at crossed-products for quantum groups, while Chapter~16 looks at the generalisation of cocycle crossed-products. It is nice to see some of this material in self-contained form, as in the existing literature, it is a little fragmented (and sometimes only explicitly worked out for Kac algebras, for example).
Chapter 17 looks at subfactor theory and links with iterated crossed-products. Chapter~18 studies Galois objects and cocycle deformations, which are linked processes for building new quantum groups out of old ones. Chapter~19 looks at double crossed products, a method of combining two quantum groups into a new one (and, along with the bicrossed product, when the starting groups are actually classical, an important source of examples of genuinely quantum groups with many unexpected properties).
The final short chapter looks at (one method of) inducing (co)representations from a quantum subgroup. Perhaps here especially, one feels the lack of a discussion of the wider literature: to give one example, it would be nice to have a comparison made with the approach of [\textit{S.~Vaes}, J. Funct. Anal. 229, No.~2, 317--374 (2005; Zbl 1087.22005)], or at least a pointer to this paper.
The reviewer must say something about the notation used in the book. In the existing literature, it is now common to write \(\mathbb G\) for a quantum group, and to denote the, say, von Neumann algebra representing it by \(L^\infty(\mathbb G)\) (even though this will in general be a non-commutative algebra, the notation is suggestive of the motivating classical case). Alternatively, the vast majority of the literature uses \(M\) to denote the von Neumann algebra, \(A\) for the \(C^*\)-algebra, \(W\) for the fundamental multiplicative unitary, \(\varphi\) for the left invariant weight. The book under review uses \(W\) for the von Neumann algebra (perhaps to evoke \(W^*\)-algebra), writes \(M\) for the multiplicative unitary, writes \(x\) for the (left) invariant weight, and so forth. While this notation is consistent, the reviewer does find that it leads to a certain cognitive load having the same letters denoting rather different objects, as compared to the existing literature.
There is an appendix, giving a basic guide to measure theory, some aspects of complex analysis used in the book, and so forth. Some exercises are set at the end of each chapter. There is a modest bibliography, listing works which are used as the basis for the material in the book, and each chapter contains a brief guide to relevant bibliographic works. However, there is no attempt to give a guide to the wider literature, or even pointers to further reading. The index is very thorough.
As previously said, this is a very ambitious book, and this review gives a sense of the volume of material covered. The necessary tradeoff is that one has to move at a fast pace, and while the book is admirably self-contained, many proofs are quite terse. This is perhaps not an introductory text. However, it forms the first textbook to really give all the details about the locally compact case of quantum groups, and is bound to be an excellent reference for researchers, as well as perhaps a good ``second course'' book for graduate students.
Reviewer: Matthew Daws (Preston)Extensions of \(C^*\)-algebras by a small idealhttps://zbmath.org/1527.460312024-02-28T19:32:02.718555Z"Lin, Huaxin"https://zbmath.org/authors/?q=ai:lin.huaxin"Ng, Ping Wong"https://zbmath.org/authors/?q=ai:ng.ping-wongSummary: We classify all essential extensions of the form
\[
0\rightarrow\mathcal{W}\rightarrow D\rightarrow A \rightarrow 0,
\]
where \(\mathcal{W}\) is the unique separable simple \(C^*\)-algebra with a unique tracial state, which is \(KK\)-contractible and has finite nuclear dimension, and \(A\) is a separable amenable \(\mathcal{W}\)-embeddable \(C^*\)-algebra, which satisfies the Universal Coefficient Theorem (UCT). We actually prove more general results. We also classify a class of amenable \(C^*\)-algebras, which have only one proper closed ideal \(\mathcal{W}\).Left regular representations of Garside categories. I: C*-algebras and groupoidshttps://zbmath.org/1527.460322024-02-28T19:32:02.718555Z"Li, Xin"https://zbmath.org/authors/?q=ai:li.xin.37|li.xin.15|li.xin.12|li.xin.1|li.xin.17|li.xin.11|li.xin.3|li.xin.18|li.xin.13|li.xin|li.xin.14Summary: We initiate the study of C*-algebras and groupoids arising from left regular representations of Garside categories, a notion which originated from the study of braid groups. Every higher rank graph is a Garside category in a natural way. We develop a general classification result for closed invariant subspaces of our groupoids as well as criteria for topological freeness and local contractiveness, properties which are relevant for the structure of the corresponding C*-algebras. Our results provide a conceptual explanation for previous results on gauge-invariant ideals of higher rank graph C*-algebras. As another application, we give a complete analysis of the ideal structures of C*-algebras generated by left regular representations of Artin-Tits monoids.A note on \(Z^\ast\) algebrashttps://zbmath.org/1527.460332024-02-28T19:32:02.718555Z"Taghavi, Ali"https://zbmath.org/authors/?q=ai:taghavi.aliSummary: We study some properties of \(Z^\ast\) algebras, those \(C^\ast\) algebras whose all positive elements are zero divisors. Using an example, we show that an extension of a \(Z^\ast\) algebra by a \(Z^\ast\) algebra is not necessarily a \(Z^\ast\) algebra. However, we prove that the extension of a non-\(Z^\ast\) algebra by a non-\(Z^\ast\) algebra is a non-\(Z^\ast\) algebra. We also prove that the tensor product of a \(Z^\ast\) algebra by a \(C^\ast\) algebra is a \(Z^\ast\) algebra. As an indirect consequence of our methods, we prove the following inequality type results: (i) Let \(a_n\) be a sequence of positive elements of a \(C^\ast\) algebra \(A\) which converges to zero. Then, there are positive sequences \(b_n\) of real numbers and \(c_n\) of elements of \(A\) which converge to zero such that \(a_{n+k}\leq b_nc_k.\) (ii) Every compact subset of the positive cones of a \(C^\ast\) algebra has an upper bound in the algebra.Permanence of real rank zero from a centrally large subalgebra to a C*-algebrahttps://zbmath.org/1527.460342024-02-28T19:32:02.718555Z"Zhao, Xia"https://zbmath.org/authors/?q=ai:zhao.xia"Fang, Xiaochun"https://zbmath.org/authors/?q=ai:fang.xiaochunSummary: Let \(A\) be an infinite dimensional simple unital C*-algebra and \(B\) be a centrally large subalgebra of \(A\). \textit{D.~E. Archey} and \textit{N.~C. Phillips} [J. Oper. Theory 83, No.~2, 353--389 (2020; Zbl 1463.46080)]
showed that \(A\) has real rank zero when \(B\) has real rank zero and stable rank one. In this paper, we have removed the restriction of \(B\) has stable rank one. Using matrix decomposition by some suitable projections, we have shown that \(A\) has real rank zero if \(B\) has real rank zero.On bounded coordinates in GNS spaceshttps://zbmath.org/1527.460352024-02-28T19:32:02.718555Z"De, Debabrata"https://zbmath.org/authors/?q=ai:de.debabrata"Mukherjee, Kunal"https://zbmath.org/authors/?q=ai:mukherjee.kunal-kSummary: We provide a comprehensive study of uniformly left bounded (resp. left-right bounded) orthonormal bases in GNS spaces of infinite-dimensional von Neumann algebras in the framework of both faithful normal states and f.n.s. weights. There are two issues to consider: one concerning the existence of such bases and the other concerning the bound in operator norm of the left (resp. left and right) multiplication operators associated to such bases. We provide necessary and sufficient conditions on a closed subspace of a GNS space to guarantee the existence of an orthonormal basis of uniformly left bounded (resp. left-right bounded) vectors. In the context of states, while a basis of the first kind exists for all GNS spaces, \(\mathbf{B}(\ell^2)\) is excluded for a basis of the latter kind. However, in the context of weights, there are no such obstructions. In the context of weights, the GNS space of every infinite-dimensional von Neumann algebra admits a uniformly left and right bounded orthonormal basis such that the aforesaid bound is arbitrarily small.
If \(M\) is an infinite-dimensional factor and \(\varphi\) is a faithful normal state on \(M\), then given \(\epsilon > 0\), the associated GNS space admits a uniformly left bounded orthonormal basis \(\mathcal{O}\) such that \(\sup_{\xi\in\mathcal{O}}\|{L_\xi}\|\leq (1+\sqrt{2})+\epsilon.\)
If \(M\), \(\varphi\) and \(\epsilon\) are as above, and \(M\) is either of type \(\mathrm{II}\) or \(\mathrm{III}_\lambda\) with \(\lambda\in [0,1)\), then the GNS space of \(\varphi\) admits a left and right bounded orthonormal basis \(\mathcal{O}\) such that
\[
\sup_{\xi\in\mathcal{O}} \max(\|{L_\xi}\|,\|{R_\xi}\|)\leq (1+\sqrt{2})+\epsilon.
\]
Similar is the conclusion if \(M\) is of type \(\mathrm{III}_1\) and \(\varphi\) is almost periodic. If \(\varphi\) is not tracial and \(\mathcal{O}\) is a uniformly left and right bounded orthonormal basis as stated above, there exists \(\delta > 0\) such that
\[
1+\delta\leq\sup_{\xi\in\mathcal{O}}\max(\|{L_\xi}\|,\|{R_\xi}\|)\leq(1+\sqrt{2})+\epsilon.
\]
Related questions on unitary bases remain open and untouched since the 1967 Baton Rouge conference.A simple nuclear \(C^{\ast}\)-algebra with an internal asymmetryhttps://zbmath.org/1527.460362024-02-28T19:32:02.718555Z"Hirshberg, Ilan"https://zbmath.org/authors/?q=ai:hirshberg.ilan"Phillips, N. Christopher"https://zbmath.org/authors/?q=ai:phillips.n-christopherThe authors construct an AH-algebra whose Elliott invariant admits an automorphism which does not lift to the AH-algebra. This example must have infinite nuclear dimension. Otherwise, the classification programme would imply that every automorphism of the Elliott invariant lifts to the \(C^{\ast}\)-algebra. The starting point for the authors is a construction of \textit{A. S. Toms} [Ann. Math. (2) 167, No. 3, 1029--1044 (2008; Zbl 1181.46047)], which in turn is based on ideas of \textit{J. Villadsen} [J. Funct. Anal. 154, No. 1, 110--116 (1998; Zbl 0915.46047)]. An important ingredient is the analysis or radii of comparison for corners.
Reviewer: Xin Li (Glasgow)Noncommutative Mulholland inequalities associated with factors and their applicationshttps://zbmath.org/1527.460372024-02-28T19:32:02.718555Z"Yang, Yongqiang"https://zbmath.org/authors/?q=ai:yang.yongqiang"Yan, Cheng"https://zbmath.org/authors/?q=ai:yan.cheng"Han, Yazhou"https://zbmath.org/authors/?q=ai:han.yazhou"Liu, Shuting"https://zbmath.org/authors/?q=ai:liu.shutingSummary: In this paper, we obtain noncommutative Mulholland inequalities associated with type I and type \(\mathrm{II}_1\) factors, which are generalizations of noncommutative Minkowski inequalities, and conclude that the noncommutative Mulholland inequalities with non-power functions fail to exist for type \(\mathrm{II}_\infty\) factors. As applications of the noncommutative Mulholland inequalities, we prove that functionals \(\phi^{-1}\circ\tau\circ\phi(|\cdot|)\) are noncommutative \(F\)-norms under same conditions of establishing the noncommutative Mulholland inequalities, and the conclusion also fails to hold for type \(\mathrm{II}_\infty\) factors if \(\phi\) are non-power functions. Furthermore, we prove that the functionals \(\phi^{-1}\circ\tau\circ\phi(|\cdot|)\) associated with type I or type II factors, are norms if and only if \(\phi(t) = \phi (1)t^p\), (\(t \geq 0\)), for some \(p \geq 1\). In addition, we define noncommutative \(F\)-normed spaces by the above noncommutative \(F\)-norms and give a positive answer about the uniform convexity of the noncommutative \(F\)-normed spaces.Duality and interpolation for symmetric Banach spaces of noncommutative quasi-martingaleshttps://zbmath.org/1527.460382024-02-28T19:32:02.718555Z"Ma, Congbian"https://zbmath.org/authors/?q=ai:ma.congbian"Fan, Liping"https://zbmath.org/authors/?q=ai:fan.liping"Zhang, Xiaoyan"https://zbmath.org/authors/?q=ai:zhang.xiaoyan.4|zhang.xiaoyan.3|zhang.xiaoyan.2"Li, Xin"https://zbmath.org/authors/?q=ai:li.xin.18|li.xin.26|li.xin.17|li.xin.3|li.xin.14|li.xin.15|li.xin.1|li.xin.11|li.xin|li.xin.37|li.xin.12|li.xin.13Summary: Let \(E\) be a symmetric Banach space with the Fatou property and \(1<p_E\le q_E<p\). We prove the duality for symmetric Banach space \(_p\widehat{E}(\mathcal{M})\) which is a kind of noncommutative quasi-martingale space. As its applications, we discuss concrete description of the symmetric Banach space \(_p\widehat{E}(\mathcal{M})\) as interpolations of quasi-martingale \(L_p\)-spaces.The Brown measure of a family of free multiplicative Brownian motionshttps://zbmath.org/1527.460392024-02-28T19:32:02.718555Z"Hall, Brian C."https://zbmath.org/authors/?q=ai:hall.brian-c"Ho, Ching-Wei"https://zbmath.org/authors/?q=ai:ho.ching-weiSummary: We consider a family of free multiplicative Brownian motions \(b_{s,\tau }\) parametrized by a real variance parameter \(s\) and a complex covariance parameter \(\tau\). We compute the Brown measure \(\mu_{s,\tau }\) of \(ub_{s,\tau },\) where \(u\) is a unitary element freely independent of \(b_{s,\tau }\). We find that \(\mu_{s,\tau }\) has a simple structure, with a density in logarithmic coordinates that is constant in the \(\tau \)-direction. These results generalize those of Driver-Hall-Kemp [\textit{B.~K. Driver} et al., J. Funct. Anal. 278, No.~1, Article ID 108303, 42~p. (2020; Zbl 1427.43008)] and Ho-Zhong [\textit{C.-W. Ho} and \textit{P.~Zhong}, J. Eur. Math. Soc. (JEMS) 25, No.~6, 2163--2227 (2023; Zbl 07714610)] for the case \(\tau =s\). We also establish a remarkable ``model deformation phenomenon,'' stating that all the Brown measures with \(s\) fixed and \(\tau\) varying are related by push-forward under a natural family of maps. Our proofs use a first-order nonlinear PDE of Hamilton-Jacobi type satisfied by the regularized log potential of the Brown measures. Although this approach is inspired by the PDE method introduced by Driver-Hall-Kemp [loc.\,cit.], our methods are substantially different at both the technical and conceptual level.Rates of convergence in the free central limit theoremhttps://zbmath.org/1527.460402024-02-28T19:32:02.718555Z"Maejima, Makoto"https://zbmath.org/authors/?q=ai:maejima.makoto"Sakuma, Noriyoshi"https://zbmath.org/authors/?q=ai:sakuma.noriyoshiSummary: We study the free central limit theorem for not necessarily identically distributed free random variables where the limiting distribution is the semicircle distribution. Starting from an estimate for the Kolmogorov distance between the measure of suitably normalized sums of free random variables and the semicircle distribution without any moment condition, we show the free Lindeberg central limit theorem and improve the known results on rates of convergence under the conditions of the existence of the third moments.Topological boundaries of covariant representationshttps://zbmath.org/1527.460412024-02-28T19:32:02.718555Z"Amini, Massoud"https://zbmath.org/authors/?q=ai:amini.massoud"Zavar, Sajad"https://zbmath.org/authors/?q=ai:zavar.sajadSummary: We associate a boundary \(\mathcal{B}_{\pi,u}\) to each covariant representation \((\pi,u,H)\) of a \(C^*\)-dynamical system \((G,A,\alpha)\) and study the action of \(G\) on \(\mathcal{B}_{\pi,u}\) and its amenability properties. We relate rigidity properties of traces on the associated crossed product \(C^*\)-algebra to faithfulness of the action of the group on this boundary.\(\mathbb{N}^k\)-actionshttps://zbmath.org/1527.460422024-02-28T19:32:02.718555Z"Farsi, Carla"https://zbmath.org/authors/?q=ai:farsi.carla"Huang, Leonard"https://zbmath.org/authors/?q=ai:huang.leonard-t"Kumjian, Alex"https://zbmath.org/authors/?q=ai:kumjian.alexander"Packer, Judith"https://zbmath.org/authors/?q=ai:packer.judith-aSummary: We consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space and study generalized Ruelle operators and \(C^{\ast}\)-algebras associated to these groupoids. We provide a new characterization of \(1\)-cocycles on these groupoids taking values in a locally compact abelian group, given in terms of \(k\)-tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle-Perron-Frobenius theory for dynamical systems of several commuting operators \((k\)-Ruelle triples and commuting Ruelle operators). Results on KMS states on \(C^{\ast}\)-algebras constructed from these groupoids are derived. When the groupoids being studied come from higher-rank graphs, our results recover existence and uniqueness results for KMS states associated to the graphs.Equivariant \({\mathcal{Z}} \)-stability for single automorphisms on simple \(C^*\)-algebras with tractable trace simpliceshttps://zbmath.org/1527.460432024-02-28T19:32:02.718555Z"Wouters, Lise"https://zbmath.org/authors/?q=ai:wouters.liseThe present paper concerns the classification of amenable group actions on classifiable C*-algebras up to cocycle conjugacy, with a focus on the conjecture that equivariant \(\mathcal{Z}\)-stability is automatic for such C*-dynamical systems. For the purposes of the paper, \(A\) is a C*-algebra that is algebraically simple, separable, nuclear and \(\mathcal{Z}\)-stable for the Jiang-Su algebra \(\mathcal {Z}\) such that its trace space \(T(A)\) is a Bauer simplex and the extremal boundary \(\partial_e T(A)\) has finite covering dimension.
Under these assumptions, it is shown that an automorphism \(\alpha\) of \(A\) is cocycle conjugate to \(\alpha \otimes \mathrm{id}_{\mathcal{Z}}\). This generalizes a recent result of Gardella-Hirshberg-Vaccaro [\textit{E. Gardella} et al., J. Math. Pures Appl. (9) 162, 76--123 (2022; Zbl 1496.46068)]. As an intermediate result, it is shown that, if \(G\) is a countable discrete group for which all finitely generated subgroups are virtually nilpotent and \(G\) acts on \(A\) by \(\alpha\) so that the induced action on \(\partial_e T(A)\) is free, then \(\alpha\) is cocycle conjugate to \(\alpha \otimes \mathrm{id}_{\mathcal{Z}}\). Furthermore, it is shown that, if \(\alpha : \mathbb{Z} \to \mathrm{Aut}(A)\) is a strongly outer action, then \(\dim_{\mathrm{Rok}}^c (\alpha) \leq 2\); as a corollary, it is shown that \(\alpha\) is cocycle conjugate to \(\alpha \otimes \delta\) for any action \(\delta : \mathbb{Z} \to \mathrm{Aut}(D)\) on a strongly self-absorbing C*-algebra \(D\) with \(A \simeq A \otimes D\).
Reviewer: Evgenios Kakariadis (Newcastle upon Tyne)Ring derivations of Murray-von Neumann algebrashttps://zbmath.org/1527.460442024-02-28T19:32:02.718555Z"Huang, Jinghao"https://zbmath.org/authors/?q=ai:huang.jinghao"Kudaybergenov, Karimbergen"https://zbmath.org/authors/?q=ai:kudaybergenov.karimbergen-k"Sukochev, Fedor"https://zbmath.org/authors/?q=ai:sukochev.fedor-aSummary: Let \(\mathcal{M}\) be a type \(\mathrm{II}_1\) von Neumann algebra, \(S(\mathcal{M})\) be the Murray-von Neumann algebra associated with \(\mathcal{M}\) and let \(\mathcal{A}\) be a \(\ast \)-subalgebra in \(S(\mathcal{M})\) with \(\mathcal{M} \subseteq \mathcal{A} \). We prove that any ring derivation \(D\) from \(\mathcal{A}\) into \(S(\mathcal{M})\) is necessarily inner. Further, we prove that if \(\mathcal{A}\) is an \(E W^\ast \)-algebra such that its bounded part \(\mathcal{A}_b\) is a \(W^\ast \)-algebra without finite type I direct summands, then any ring derivation \(D\) from \(\mathcal{A}\) into \(L S( \mathcal{A}_b)\) -- the algebra of all locally measurable operators affiliated with \(\mathcal{A}_b\), is an inner derivation. We also give an example showing that the condition \(\mathcal{M} \subseteq \mathcal{A}\) is essential. At the end of this paper, we provide several criteria for an abelian extended \(W^\ast \)-algebra such that all ring derivations on it are linear.Dunkl operators for arbitrary finite groupshttps://zbmath.org/1527.460452024-02-28T19:32:02.718555Z"Đurđevich, Micho"https://zbmath.org/authors/?q=ai:durdevich.micho"Sontz, Stephen Bruce"https://zbmath.org/authors/?q=ai:sontz.stephen-bruceSummary: The Dunkl operators associated with a necessarily finite Coxeter group acting on a Euclidean space are generalized to any finite group using the techniques of non-commutative geometry, as introduced by the authors to view the usual Dunkl operators as covariant derivatives in a quantum principal bundle with a quantum connection. The definitions of Dunkl operators and their corresponding Dunkl connections are generalized to quantum principal bundles over quantum spaces which possess a classical finite structure group. We introduce cyclic Dunkl connections and their cyclic Dunkl operators. Then, we establish a number of interesting properties of these structures, including the characteristic zero-curvature property. Particular attention is given to the example of complex reflection groups, and their naturally generalized siblings called groups of Coxeter type.Dirac operators for matrix algebras converging to coadjoint orbitshttps://zbmath.org/1527.460462024-02-28T19:32:02.718555Z"Rieffel, Marc A."https://zbmath.org/authors/?q=ai:rieffel.marc-aSummary: In the high-energy physics literature one finds statements such as ``matrix algebras converge to the sphere''. Earlier I provided a general precise setting for understanding such statements, in which the matrix algebras are viewed as quantum metric spaces, and convergence is with respect to a quantum Gromov-Hausdorff-type distance. But physicists want even more to treat structures on spheres (and other spaces), such as vector bundles, Yang-Mills functionals, Dirac operators, etc., and they want to approximate these by corresponding structures on matrix algebras. In the present paper we provide a somewhat unified construction of Dirac operators on coadjoint orbits and on the matrix algebras that converge to them. This enables us to prove our main theorem, whose content is that, for the quantum metric-space structures determined by the Dirac operators that we construct, the matrix algebras do indeed converge to the coadjoint orbits, for a quite strong version of quantum Gromov-Hausdorff distance.Callias-type operators associated to spectral tripleshttps://zbmath.org/1527.460472024-02-28T19:32:02.718555Z"Schulz-Baldes, Hermann"https://zbmath.org/authors/?q=ai:schulz-baldes.hermann"Stoiber, Tom"https://zbmath.org/authors/?q=ai:stoiber.tomThe Callias index theorem and its even dimensional analogue give formulas for the index of Dirac operators on non-compact manifolds which are perturbed by self-adjoint potentials that act on sections of a finite-dimensional vector bundle and are invertible at infinity. There are many possible generalizations, for example, one can allow infinite-dimensional vector bundles as in the Robbin-Salamon theorem [\textit{J.~Robbin} and \textit{D.~Salamon}, Bull. Lond. Math. Soc. 27, No.~1, 1--33 (1995; Zbl 0859.58025)] or Hilbert-module bundles of finite type [\textit{M.~Braverman} and \textit{S.~Cecchini}, J. Geom. Anal. 28, No.~1, 546--586 (2018; Zbl 1390.19009); \textit{S.~Cecchini}, J. Topol. Anal. 12, No.~4, 897--939 (2020; Zbl 1451.19019)].
In the present paper, the authors generalize in a different direction, namely, the underlying manifold is replaced by a semifinite spectral triple and the perturbing potentials are elements of a certain multiplier algebra. In this generalized setting, the authors are still able to express the index of a Callias-type operator in terms of an index pairing derived from the spectral triple. The precise statement is given in Section~3 (Theorem~8) and proved in Section~4, for the case of unbounded potentials in Section~6 (Theorem~31). Section~5 states and proves an even analogue for pairings of an even spectral triple with a potential having a further symmetry (Theorem~17). Section~6 then also covers the unbounded even case (Theorem~37). Section~7 outlines how their Callias index arises as an unbounded representative of a \(KK\)-group. Section~8 presents classical and new examples. Finally, it is noteworthy that, following [\textit{J.~Kaad} and \textit{M.~Lesch}, Adv. Math. 248, 495--530 (2013; Zbl 1294.19001); \textit{K.~van den Dungen}, J. Spectr. Theory 9, No.~4, 1459--1506 (2019; Zbl 1439.19008)], the authors interpret their index pairing as a non-commutative analogue of spectral flow.
Reviewer: Stefan Wagner (Karlskrona)Equivariant homologies for operator algebras; a surveyhttps://zbmath.org/1527.460482024-02-28T19:32:02.718555Z"Shirinkalam, A."https://zbmath.org/authors/?q=ai:shirinkalam.ahmadSummary: This is a survey of a variety of equivariant (co)homology theories for operator algebras. We briefly discuss a background on equivariant Hochschild cohomology. We discuss a notion of equivariant \(L^2\)-cohomology and equivariant \(L^2\)-Betti numbers for subalgebras of a von Neumann algebra. For graded \(C^*\)-algebras (with grading over a group) we elaborate on a notion of graded \(L^2\)-cohomology and its relation to equivariant \(L^2\)-cohomology.Solvability of Infinite systems of differential equations of general order in the sequence space \(bv_\infty\)https://zbmath.org/1527.470142024-02-28T19:32:02.718555Z"Saboori, M. H."https://zbmath.org/authors/?q=ai:saboori.mohammad-hassan"Hassani, M."https://zbmath.org/authors/?q=ai:hassani.mahmoud"Allahyari, R."https://zbmath.org/authors/?q=ai:allahyari.rezaSummary: We introduce the Hausdorff measure of noncompactness in the sequence space \(bv_\infty\) and investigate the existence of solution of infinite systems of differential equations with respect to Hausdorff measure of noncompactness. Finally, we present an example to defend of theorem of existential.A note on the equivalence and the boundary behavior of a class of Sobolev capacitieshttps://zbmath.org/1527.490072024-02-28T19:32:02.718555Z"Christof, Constantin"https://zbmath.org/authors/?q=ai:christof.constantin"Müller, Georg"https://zbmath.org/authors/?q=ai:muller.georgSummary: The purpose of this paper is to study different notions of Sobolev capacity commonly used in the analysis of obstacle- and Signorini-type variational inequalities. We review basic facts from capacity theory in an abstract setting that is tailored to the study of \(W^{1,p}\)- and \(W^{1-1}/^{p,p}\)-capacities, and we prove equivalency results that relate several approaches found in the literature to each other. Motivated by applications in contact mechanics, we especially focus on the behavior of different Sobolev capacities on and near the boundary of the domain in question. As a result, we obtain, for example, that the most common approaches to the sensitivity analysis of Signorini-type problems are exactly the same.Best proximity point for \(q\)-ordered proximal contraction in noncommutative Banach spaceshttps://zbmath.org/1527.540222024-02-28T19:32:02.718555Z"Bartwal, Ayush"https://zbmath.org/authors/?q=ai:bartwal.ayush"Rawat, Shivam"https://zbmath.org/authors/?q=ai:rawat.shivam"Beg, Ismat"https://zbmath.org/authors/?q=ai:beg.ismatSummary: We introduce the concept of \(q\)-ordered proximal nonunique contraction for the non self mappings and then obtain some proximity point results for these mappings. We also furnish examples to support our claims.Matrix concentration inequalities and free probabilityhttps://zbmath.org/1527.600202024-02-28T19:32:02.718555Z"Bandeira, Afonso S."https://zbmath.org/authors/?q=ai:bandeira.afonso-s"Boedihardjo, March T."https://zbmath.org/authors/?q=ai:boedihardjo.march-t"van Handel, Ramon"https://zbmath.org/authors/?q=ai:van-handel.ramonLet \(X\) be a \(d\times d\) Gaussian random matrix. The authors establish non-asymptotic bounds on the spectrum of such a random matrix \(X\) with improved behaviour in terms of the dimension \(d\) in the non-commutative case compared to the usual Khintchine inequality. These bounds reflect the extent to which the spectrum of \(X\) differs from that of a corresponding model \(X_{\text{free}}\) arising in free probability, and make explicit the cases in which these two spectra are the same to leading order. They further lead to bounds on various spectral statistics of \(X\) in terms of \(X_{\text{free}}\), and linearization allows the authors to establish strong asymptotic freeness for a large class of random matrices. A universality principle extends results on Gaussian matrices to the case of sums of independent random matrices.
Reviewer: Fraser Daly (Edinburgh)Thermal equilibrium distribution in infinite-dimensional Hilbert spaceshttps://zbmath.org/1527.600282024-02-28T19:32:02.718555Z"Tumulka, Roderich"https://zbmath.org/authors/?q=ai:tumulka.roderichSummary: The thermal equilibrium distribution over quantum-mechanical wave functions is a so-called Gaussian adjusted projected (GAP) measure, \(\text{GAP}(\rho_\beta)\), for a thermal density operator \(\rho_\beta\) at inverse temperature \(\beta\). More generally, \(\text{GAP}(\rho)\) is a probability measure on the unit sphere in Hilbert space for any density operator \(\rho\) (i.e. a positive operator with trace 1). In this note, we collect the mathematical details concerning the rigorous definition of \(\text{GAP}(\rho)\) in infinite-dimensional separable Hilbert spaces. Its existence and uniqueness follows from Prokhorov's theorem on the existence and uniqueness of Gaussian measures in Hilbert spaces with given mean and covariance. We also give an alternative existence proof. Finally, we give a proof that \(\text{GAP}(\rho)\) depends continuously on \(\rho\) in the sense that convergence of \(\rho\) in the trace norm implies weak convergence of \(\text{GAP}(\rho)\).Double-graded quantum superplanehttps://zbmath.org/1527.810812024-02-28T19:32:02.718555Z"Bruce, Andrew James"https://zbmath.org/authors/?q=ai:bruce.andrew-james"Duplij, Steven"https://zbmath.org/authors/?q=ai:duplij.stevenSummary: A \(\mathbb{Z}_2\times\mathbb{Z}_2\)-graded generalisation of the quantum superplane is proposed and studied. We construct a bicovariant calculus on what we shall refer to as the \textit{double-graded quantum superplane}. The commutation rules between the coordinates, their differentials and partial derivatives are explicitly given. Furthermore, we show that an extended version of the double-graded quantum superplane admits a natural Hopf \(\mathbb{Z}_2^2\)-algebra structure.\(W^\star\) dynamics of infinite dissipative quantum systemshttps://zbmath.org/1527.810902024-02-28T19:32:02.718555Z"Sewell, Geoffrey L."https://zbmath.org/authors/?q=ai:sewell.geoffrey-lSummary: We formulate the dynamics of an infinitely extended open dissipative quantum system, \(\Sigma\), in the Schrödinger picture. The generic model on which this is based comprises a \(C^\star\)-algebra, \(\mathcal{A}\), of observables, a folium, \(\mathcal{F}\), of states on this algebra and a one-parameter semigroup, \(\tau\), of linear transformations of \(\mathcal{F}\) that represents its dynamics and is given by a natural infinite-volume limit of the corresponding semigroup for a finite system. On this basis, we establish that the dynamics of \(\Sigma\) is given by a one-parameter group of completely positive linear transformations of the \(W^\star\)-algebra dual to \(\mathcal{F}\). This result serves to extend our earlier formulation [\textit{G. L. Sewell}, Lett. Math. Phys. 6, 209--213 (1982; Zbl 0541.46058)] of infinitely extended conservative systems to open dissipative ones.