Recent zbMATH articles in MSC 46L https://zbmath.org/atom/cc/46L 2022-06-24T15:10:38.853281Z Werkzeug Troupes, cumulants, and stack-sorting https://zbmath.org/1485.05004 2022-06-24T15:10:38.853281Z "Defant, Colin" https://zbmath.org/authors/?q=ai:defant.colin Summary: In several cases, a sequence of free cumulants that counts certain binary plane trees corresponds to a sequence of classical cumulants that counts the decreasing versions of the same trees. Using two new operations on binary plane trees that we call insertion and decomposition, we prove that this surprising phenomenon holds for families of trees that we call troupes. We give a simple characterization of troupes, showing that they are plentiful. Troupes provide a broad framework for generalizing several of the results that are known about West's stack-sorting map $$s$$. Indeed, we give new proofs of some of the main theorems underlying techniques that have been developed recently for understanding $$s$$; these new proofs are far more conceptual than the original ones, explain how the objects called valid hook configurations arise very naturally, and generalize to the context of troupes. To illustrate these general techniques, we enumerate 2-stack-sortable and 3-stack-sortable alternating permutations of odd length and 2-stack-sortable and 3-stack-sortable permutations whose descents are all peaks. The unexpected connection between troupes and cumulants provides a powerful new tool for analyzing the stack-sorting map that hinges on free probability theory. We give numerous applications of this method. For example, we show that if $$\sigma \in S_{n - 1}$$ is chosen uniformly at random and des denotes the descent statistic, then the expected value of $$\operatorname{des}(s(\sigma)) + 1$$ is $\left(3 - \sum_{j = 0}^n \frac{ 1}{ j !}\right) n.$ Furthermore, the variance of $$\operatorname{des}(s(\sigma)) + 1$$ is asymptotically $$(2 + 2 e - e^2) n$$. We obtain similar results concerning the expected number of descents of postorder readings of decreasing binary plane trees of various types. We also obtain improved estimates for $$|s(S_n)|$$ and an improved lower bound for the degree of noninvertibility of $$s : S_n \to S_n$$. The combinatorics of valid hook configurations allows us to give two novel formulas that convert from free to classical (univariate) cumulants. The first formula is given by a sum over noncrossing partitions, and the second is given by a sum over 231-avoiding valid hook configurations. We pose several conjectures and open problems. On nonlinear random approximation of 3-variable Cauchy functional equation https://zbmath.org/1485.39036 2022-06-24T15:10:38.853281Z "Cho, Yeol Je" https://zbmath.org/authors/?q=ai:cho.yeol-je "Kang, Shin Min" https://zbmath.org/authors/?q=ai:kang.shin-min "Rassias, Themistocles M." https://zbmath.org/authors/?q=ai:rassias.themistocles-m "Saadati, Reza" https://zbmath.org/authors/?q=ai:saadati.reza Summary: In this paper, we study to approximate the homomorphisms and derivations for 3-variable Cauchy functional equations in $$RC^\ast$$-algebras and Lie $$RC^\ast$$-algebras by the fixed point method. Quantum Fourier analysis https://zbmath.org/1485.42013 2022-06-24T15:10:38.853281Z "Jaffe, Arthur" https://zbmath.org/authors/?q=ai:jaffe.arthur-m "Jiang, Chunlan" https://zbmath.org/authors/?q=ai:jiang.chunlan "Wu, Jinsong" https://zbmath.org/authors/?q=ai:wu.jinsong Summary: Quantum Fourier analysis is a subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory) with analytic estimates. This provides interesting tools to investigate phenomena such as quantum symmetry. We establish bounds on the quantum Fourier transform, as a map between suitably defined $$L^p$$ spaces, leading to an uncertainty principle for relative entropy. We cite several applications of quantum Fourier analysis in subfactor theory, in category theory, and in quantum information. We suggest a topological inequality, and we outline several open problems. Some relatives of the Catalan sequence https://zbmath.org/1485.44005 2022-06-24T15:10:38.853281Z "Liszewska, Elżbieta" https://zbmath.org/authors/?q=ai:liszewska.elzbieta "Młotkowski, Wojciech" https://zbmath.org/authors/?q=ai:mlotkowski.wojciech Summary: We study a family of sequences $$c_n( a_2,\dots,a_r)$$, where $$r \geq 2$$ and $$a_2,\dots,a_r$$ are real parameters. We find a sufficient condition for positive definiteness of the sequence $$c_n( a_2,\dots,a_r)$$ and check several examples from OEIS. We also study relations of these sequences with the free and monotone convolution. On $$C^*$$-algebras generated by the set of probability distributions https://zbmath.org/1485.46053 2022-06-24T15:10:38.853281Z "Amosov, G. G." https://zbmath.org/authors/?q=ai:amosov.grigorii-gennadevich "Grigoryan, S. A." https://zbmath.org/authors/?q=ai:grigoryan.suren-a "Kuznetsova, A. Yu." https://zbmath.org/authors/?q=ai:kuznetsova.alla-yu Summary: In this paper, we consider $$C^*$$-algebras generated by Markov operators induced by a probabilistic dynamical system. It is shown that in some cases these algebras are isomorphic to Toeplitz, Cuntz, and Cuntz-Toeplitz algebras. By the probabilistic dynamic system, the Markov operator is constructed, and it is shown that it can be extended to a quantum channel on a convex set of positive operators with unit trace. Stability of rotation relation of two unitaries with the flip action in $$C^\ast$$-algebras https://zbmath.org/1485.46054 2022-06-24T15:10:38.853281Z "Hua, Jiajie" https://zbmath.org/authors/?q=ai:hua.jiajie Summary: We show that if $$\theta \in(0, 1)$$ is an irrational number, then for any $$\varepsilon > 0$$, there exists $$\delta > 0$$ satisfying the following: For any unital $$C^\ast$$-algebra $$A$$ with the cancellation property, strict comparison and nonempty tracial state space, any three unitaries $$u, v, w \in A$$ such that (1) $$\| u v - e^{2 \pi i \theta} v u \| < \delta$$, $$w u w^{- 1} = u^{- 1}$$, $$w v w^{- 1} = v^{- 1}$$, $$w^2 = 1_A$$, (2) $$\tau(a w) = 0$$ and $$\tau( ( u v u^\ast v^\ast )^n) = e^{2 \pi i n \theta}$$ for all $$n \in \mathbb{N}$$, all $$a \in C^\ast(u, v)$$ and all tracial state $$\tau$$ on $$A$$, where $$C^\ast(u, v)$$ is the $$C^\ast$$-subalgebra generated by $$u$$ and $$v$$, there exists a triple of unitaries $$\widetilde{u}, \widetilde{v}, \widetilde{w} \in A$$ such that \begin{align*} \widetilde{u} \widetilde{v} &= e^{2 \pi i \theta} \widetilde{v} \widetilde{u}, \quad \widetilde{w} \widetilde{u} \widetilde{w}^{- 1} = \widetilde{u}^{- 1}, \quad \widetilde{w} \widetilde{v} \widetilde{w}^{- 1} = \widetilde{v}^{- 1}, \quad \widetilde{w}^2 = 1_A \quad\text{ and} \\ \| u - \widetilde{u} \| &< \varepsilon, \quad \| v - \widetilde{v} \| < \varepsilon, \quad \| w - \widetilde{w} \| < \varepsilon . \end{align*} Towards a sheaf cohomology theory for $$C^\ast$$-algebras https://zbmath.org/1485.46055 2022-06-24T15:10:38.853281Z "Mathieu, Martin" https://zbmath.org/authors/?q=ai:mathieu.martin Summary: In joint work with Pere Ara (Barcelona) we are in the process of developing a full sheaf cohomology theory for noncommutative $$C^\ast$$-algebras. In this survey, we discuss the difficulties arising from the fact that the appropriate categories of operator module sheaves over sheaves of $$C^\ast$$-algebras are non-abelian and therefore the homology theory needed has to be set in the more general framework of exact categories. For the entire collection see [Zbl 1448.46007]. Ternary operator categories https://zbmath.org/1485.46056 2022-06-24T15:10:38.853281Z "Pluta, Robert" https://zbmath.org/authors/?q=ai:pluta.robert "Russo, Bernard" https://zbmath.org/authors/?q=ai:russo.bernard Summary: $$\operatorname{T}^\ast$$-categories are introduced as a ternary generalization of $$\operatorname{C}^\ast$$-categories. Their linking $$\operatorname{C}^\ast$$-categories are constructed and the Gelfand-Naimark representation theorems of Zettl for $$\operatorname{C}^\ast$$-ternary rings and for $$\operatorname{W}^\ast$$-ternary rings, are generalized to $$\operatorname{T}^\ast$$-categories. Biduals of $$\operatorname{C}^\ast$$-categories and of $$\operatorname{T}^\ast$$-categories are considered. Generalized inductive limits and maximal stably finite quotients of $$C^\ast$$-algebras https://zbmath.org/1485.46057 2022-06-24T15:10:38.853281Z "Yao, Hongliang" https://zbmath.org/authors/?q=ai:yao.hongliang.1 Summary: For any $$C^\ast$$-algebra $$A$$, there is the smallest ideal $$I(A)$$ of $$A$$ such that the quotient $$A / I(A)$$ is stably finite. Let $$\varphi$$ be a $$^\ast$$-homomorphism from a $$C^\ast$$-algebra $$A$$ to a $$C^\ast$$-algebra $$B$$. It is obvious that $$\phi(I(A)) \subset I(B)$$. We denote the restriction $$\varphi$$ to $$I(A)$$ by $$I(\phi)$$. Then $$I$$ is a functor between categories of $$C^\ast$$-algebras. In this paper, we will consider exactness of the functor $$I$$. For this purpose, we give the definition of a generalized inductive system of a directed set of $$C^\ast$$-algebras and some general results about such limits. Minimal Hermitian compact operators related to a $$\mathrm{C}^\ast$$-subalgebra of $$K(H)$$ https://zbmath.org/1485.46058 2022-06-24T15:10:38.853281Z "Zhang, Ying" https://zbmath.org/authors/?q=ai:zhang.ying.1|zhang.ying.4|zhang.ying.2|zhang.ying|zhang.ying.5|zhang.ying.3 "L. N. Jiang, Lining" https://zbmath.org/authors/?q=ai:l-n-jiang.lining Summary: Let $$K(H)$$ be the $$\mathrm{C}^\ast$$-algebra of compact operators on a separable Hilbert space $$H$$. This paper studies the properties of Hermitian compact operators $$Y$$ such that $\| Y \| \leq \| Y + W \| \text{ for all } W \in W(H),$ where $$W(H)$$ is a $$\mathrm{C}^\ast$$-subalgebra of $$K(H)$$. Such a $$Y$$ is called minimal related to $$W(H)$$. The necessary and sufficient conditions that are required for $$Y$$ to be minimal related to $$W(H)$$ are characterized. Moreover, a particular $$\mathrm{C}^\ast$$-subalgebra $$W(H)$$ such that there is a quasi-conditional expectation $$E$$ from $$K(H)$$ onto it is considered, and several examples are provided. Projective and free matricially normed spaces https://zbmath.org/1485.46059 2022-06-24T15:10:38.853281Z "Helemskii, A. Ya." https://zbmath.org/authors/?q=ai:helemskii.alexander-ya Summary: We introduce and study metrically projective and metrically free matricially normed spaces. We describe these spaces in terms of a special space $$\widehat M_n$$, the space of $$n\times n$$ matrices, endowed with a special matrix-norm. We show that metrically free matricially normed spaces are matricial $$\ell_1$$-sums of some distinguished families of matricially normed spaces $$\widehat M_n$$, whereas metrically projective matricially normed spaces are complete direct summands of matricial $$\ell_1$$-sums of arbitrary families of spaces $$\widehat M_n$$. At the end we specify the underlying normed space of $$\widehat M_n$$ and show that the spaces $$\widehat M_n$$ do not belong to any of the classes $$L^p$$; $$p\in[1,\infty]$$, introduced by Effros and Ruan. However, in a certain sense the behavior of $$\widehat M_n$$ resembles that of $$L^1$$-spaces. For the entire collection see [Zbl 1448.46007]. Left multipliers of reproducing kernel Hilbert $$C^\ast$$-modules and the Papadakis theorem https://zbmath.org/1485.46060 2022-06-24T15:10:38.853281Z "Ghaemi, Mostafa" https://zbmath.org/authors/?q=ai:ghaemi.mostafa "Manuilov, Vladimir M." https://zbmath.org/authors/?q=ai:manuilov.v-m "Moslehian, Mohammad Sal" https://zbmath.org/authors/?q=ai:moslehian.mohammad-sal Summary: We give a modified definition of a reproducing kernel Hilbert $$C^\ast$$-module (shortly, $$R K H C^\ast M$$) without using the condition of self-duality and discuss some related aspects; in particular, an interpolation theorem is presented. We investigate the exterior tensor product of $$R K H C^\ast M$$ s and find their reproducing kernel. In addition, we deal with left multipliers of $$R K H C^\ast M$$s. Under some mild conditions, it is shown that one can make a new $$R K H C^\ast M$$ via a left multiplier. Moreover, we introduce the Berezin transform of an operator in the context of $$R K H C^\ast M$$ s and construct a unital subalgebra of the unital $$C^\ast$$-algebra consisting of adjointable maps on an $$R K H C^\ast M$$ and show that it is closed with respect to a certain topology. Finally, the Papadakis theorem is extended to the setting of $$R K H C^\ast M$$, and in order for the multiplication of two specific functions to be in the Papadakis $$R K H C^\ast M$$, some conditions are explored. Invariant complementation property and fixed point sets on power bounded elements in the group von Neumann algebra https://zbmath.org/1485.46061 2022-06-24T15:10:38.853281Z "Lau, Anthony To-Ming" https://zbmath.org/authors/?q=ai:lau.anthony-to-ming Summary: In this paper, we discuss the invariant complementation property of the group von Neumann algebra VN$$(G)$$ generated by the left translation operators on $$L^2(G)$$ of a locally compact group $$G$$, and the fixed point set of power bounded elements in VN$$(G)$$. For the entire collection see [Zbl 1448.46007]. Logarithmic submajorizations inequalities for operators in a finite von Neumann algebra https://zbmath.org/1485.46062 2022-06-24T15:10:38.853281Z "Yan, Cheng" https://zbmath.org/authors/?q=ai:yan.cheng "Han, Yazhou" https://zbmath.org/authors/?q=ai:han.yazhou Summary: The aim of this paper is to study the logarithmic submajorizations inequalities for operators in a finite von Neumann algebra. Firstly, some logarithmic submajorizations inequalities due to Garg and Aulja are extended to the case of operators in a finite von Neumann algebra. As an application, we get some new Fuglede-Kadison determinant inequalities of operators in that circumstance. Secondly, we improve and generalize to the setting of finite von Neumann algebras, a generalized Hölder type generalized singular numbers inequality. A non-diagonalizable pure state https://zbmath.org/1485.46063 2022-06-24T15:10:38.853281Z "Koszmider, Piotr" https://zbmath.org/authors/?q=ai:koszmider.piotr-b Summary: The paper provides a new construction of a pure state on the algebra of all bounded linear operators on the infinite-dimensional Hilbert space. Such pure states correspond to physical pure states of quantum systems with infinitely many degrees of freedom. The special property of the state is that it cannot be reduced to a pure state on any of the algebras of all diagonal operators with respect to some orthonormal basis (cannot be diagonalized). This solves in the negative a 40-y-old conjecture of J. Anderson without any extra set-theoretic hypothesis, unlike the previous results of this type. Characterizing traces on crossed products of noncommutative $$\mathrm{C}^\ast$$-algebras https://zbmath.org/1485.46064 2022-06-24T15:10:38.853281Z "Ursu, Dan" https://zbmath.org/authors/?q=ai:ursu.dan Summary: We give complete descriptions of the tracial states on both the universal and reduced crossed products of a $$\mathrm{C}^\ast$$-dynamical system consisting of a unital $$\mathrm{C}^\ast$$-algebra and a discrete group. In particular, we also answer the question of when the tracial states are in canonical bijection with the invariant tracial states on the original $$\mathrm{C}^\ast$$-algebra. This generalizes the unique trace property for discrete groups. The analysis simplifies greatly in various cases, for example when the conjugacy classes of the original group are all finite, and in other cases gives previously known results, for example when the original $$\mathrm{C}^\ast$$-algebra is commutative. We also obtain results and examples in the case of abelian groups that contradict existing results in the literature of Bédos and Thomsen. Specifically, we give a finite-dimensional counterexample, and provide a correction to the result of Thomsen. $$\mathcal{Z}$$-stability of transformation group $$\mathrm{C}^*$$-algebras https://zbmath.org/1485.46065 2022-06-24T15:10:38.853281Z "Niu, Zhuang" https://zbmath.org/authors/?q=ai:niu.zhuang The main result of this paper states that, given a free and minimal topological dynamical system with amenable acting group and compact Hausdorff underlying space, which has the uniform Rokhlin property and Cuntz comparison of open sets, mean dimension zero implies that the crossed product of the dynamical system tensorially absorbs the Jiang-Su algebra. The latter is significant because it implies that the crossed product is classified by its Elliott invariant, which in turn is an application of advances in the Elliott classification programme for regular C*-algebras. In addition, it is shown that the two crucial properties required for the main result are satisfied for actions of free abelian groups of finite rank and for dynamical systems which are extensions of Cantor systems whose acting groups have subexponential growth. Reviewer: Xin Li (Glasgow) A revised augmented Cuntz semigroup https://zbmath.org/1485.46066 2022-06-24T15:10:38.853281Z "Robert, Leonel" https://zbmath.org/authors/?q=ai:robert.leonel "Santiago, Luis" https://zbmath.org/authors/?q=ai:santiago.luis Summary: We revise the construction of the augmented Cuntz semigroup functor used by the first author to classify inductive limits of $$1$$-dimensional noncommutative CW complexes. The original construction has good functorial properties when restricted to the class of C*-algebras of stable rank one. The construction proposed here has good properties for all C*-algebras: we show that the augmented Cuntz semigroup is a stable, continuous, split exact functor, from the category of C*-algebras to the category of Cu-semigroups. On the cyclic automorphism of the Cuntz algebra and its fixed-point algebra https://zbmath.org/1485.46067 2022-06-24T15:10:38.853281Z "Aiello, Valeriano" https://zbmath.org/authors/?q=ai:aiello.valeriano "Rossi, Stefano" https://zbmath.org/authors/?q=ai:rossi.stefano Summary: We investigate the structure of the fixed-point algebra of $$\mathcal{O}_n$$ under the action of the cyclic permutation of the generating isometries. We prove that it is $$\ast$$-isomorphic with $$\mathcal{O}_n$$, thus generalizing a result of Choi and Latrémolière on $$\mathcal{O}_2$$. As an application of the technique employed, we also describe the fixed-point algebra of $$\mathcal{O}_{2 n}$$ under the exchange automorphism. Forcing axioms and coronas of $$\mathrm{C}^{\ast}$$-algebras https://zbmath.org/1485.46068 2022-06-24T15:10:38.853281Z "McKenney, Paul" https://zbmath.org/authors/?q=ai:mckenney.paul-e "Vignati, Alessandro" https://zbmath.org/authors/?q=ai:vignati.alessandro $$M$$-embedded symmetric operator spaces and the derivation problem https://zbmath.org/1485.46069 2022-06-24T15:10:38.853281Z "Huang, Jinghao" https://zbmath.org/authors/?q=ai:huang.jinghao "Levitina, Galina" https://zbmath.org/authors/?q=ai:levitina.galina "Sukochev, Fedor" https://zbmath.org/authors/?q=ai:sukochev.fedor-a A subspace $$Y$$ of a Banach space $$X$$ is called an $$M$$-ideal if there exists a projection $$P$$ from $$X^*$$ onto $$Y^\perp=\{x^*\in X^*\colon x^*|Y=0\}$$ such that for every $$x^*\in X^*$$ one has $$\|x^*\|=\|Px^*\|+\|x^*-Px^*\|$$. A space is $$M$$-embedded if it is an $$M$$-ideal of its bidual. Let $$\mathcal{M}$$ be a semifinite von Neumann algebra with a faithful semifinite normal trace $$\tau.$$ The authors prove that if $$E(0,\infty)$$ is an $$M$$-embedded fully symmetric function space with order continuous norm which does not contain the space of essentially bounded functions on $$(0,\infty)$$ that vanish at infinity, then the noncommutative function space $$E(\mathcal{M},\tau)$$ is $$M$$-embedded as well. It is shown how to apply the theorem to the derivation problems. In particular, if $$\delta: \mathcal{A}\to \mathcal{L}^{p,1}(\mathcal{M},\tau),\,1<p<\infty,$$ is a derivation from a $$C^*$$-subalgebra of $$\mathcal{M}$$ into the noncommutative Lorentz space $$\mathcal{L}^{p,1}$$, then there is an element $$T$$ of the space such that $$\delta=\delta_T$$ on $$\mathcal{A}$$ and $$\|T\|_{\mathcal{L}^{p,1}}\leq \|\delta\|_{\mathcal{A}\to\mathcal{L}^{p,1}}$$. Reviewer: Stanisław Goldstein (Łódź) Alberti-Uhlmann problem on Hardy-Littlewood-Pólya majorization https://zbmath.org/1485.46070 2022-06-24T15:10:38.853281Z "Huang, J." https://zbmath.org/authors/?q=ai:huang.jinghao "Sukochev, F." https://zbmath.org/authors/?q=ai:sukochev.fedor-a Let $$\mathcal{M}$$ be a semifinite von Neumann algebra with a faithful semifinite normal trace $$\tau.$$ The authors solve a series of open problems centered around the following question: does, for each $$y\in L^1(\mathcal{M},\tau)$$, the doubly stochastic orbit of $$y$$ coincide with its orbit in the sense of Hardy-Littlewood-Pólya? To be more specific: For $$x,y\in L^1(\mathcal{M},\tau)$$ we say that $$x$$ is \textit{majorized} by $$y$$ in the sense of Hardy-Littlewood-Pólya (denoted by $$x \prec y$$) if $$x_+ \prec\prec y_+,\, x_- \prec\prec y_-$$ and $$\tau(x) = \tau(y)$$, where $$x\prec\prec y$$ means in turn that $$\int_{0}^{t}\mu(s;x)ds\leq \int_{0}^{t}\mu(s;y)ds$$ for all $$t\geq0$$, with $$\mu$$ being the generalized singular value function. A positive linear map $$\phi$$ is called doubly stochastic if it preserves both the unit of $$\mathcal{M}$$ and its trace $$\tau$$. It turns out that whenever $$\mathcal{M}$$ is finite (and the trace is finite) or non-$$\sigma$$-finite, then for each pair $$x\prec y$$ one can find a doubly stochastic map $$\phi$$ such that $$\phi(y)=x$$. On the other hand, if the algebra $$\mathcal{M}$$ is $$\sigma$$-finite, but the trace is infinite, then the result fails. Nevertheless, one can still find such a doubly stochastic matrix in a von Neumann algebra containing $$\mathcal{M}$$ as its subalgebra. Another important result characterizes extreme points of the set of all elements of $$L^1(\mathcal{M},\tau)_h$$ majorized by some $$y\in L^1(\mathcal{M},\tau)_h$$. Reviewer: Stanisław Goldstein (Łódź) Two-parameter $$\sigma$$-$$C^*$$-dynamical systems and application https://zbmath.org/1485.46071 2022-06-24T15:10:38.853281Z "Mosadeq, M." https://zbmath.org/authors/?q=ai:mosadeq.maysam (no abstract) Local triple derivations from C*-algebras into their iterated duals https://zbmath.org/1485.46072 2022-06-24T15:10:38.853281Z "Niazi, Mohsen" https://zbmath.org/authors/?q=ai:niazi.mohsen "Miri, Mohammad Reza" https://zbmath.org/authors/?q=ai:miri.mohammad-reza (no abstract) On the Baum-Connes conjecture for discrete quantum groups with torsion and the quantum Rosenberg conjecture https://zbmath.org/1485.46073 2022-06-24T15:10:38.853281Z "Arano, Yuki" https://zbmath.org/authors/?q=ai:arano.yuki "Skalski, Adam" https://zbmath.org/authors/?q=ai:skalski.adam-g The paper under review studies the Baum-Connes conjecture for discrete quantum groups with possible torsion. Let $$\mathbf{G}$$ be a compact quantum group in the sense of Woronowicz and $$\mathbf{ {\Gamma}}$$ be the discrete dual. The main technical result in this paper is a decomposition of the equivariant category $$KK^{\mathbb{\mathbf{G}}}$$ in terms of crossed products and cofibrant objects (i.e., $$\mathbf{G}$$-C*-algebras of the form $$D\otimes A$$ the action of the form $$\alpha\otimes\mathrm{id}$$, where $$D$$ is finite-dimensional equipped with a $${\mathbf{G}}$$-action $$\alpha$$). This decomposition procedure behaves well with the Universal Coefficient Theorem (UCT in short) of C*-algebras. Indeed, the authors use this decomposition to prove that if $$\mathbf{{\Gamma}}$$ satisfies the corresponding cofibrant version of Baum-Connes property, then the UCT is preserved by the crossed product by $$\mathbf{\Gamma}$$. In particular, the reduced group C*-algebra $$C(\mathbb{\mathbf{G}})$$ for such $$\mathbb{\mathbf{\Gamma}}$$ always satisfies the UCT. Let us remark that the aforementioned cofibrant version of Baum-Connes property has been established for quite a lot of discrete quantum groups, thus we obtain various examples of quantum group C*-algebras with the UCT. As another application of this method, the authors also discuss the quantum Rosenberg conjecture, i.e., the quasidiagonality of reduced quantum group C*-algebras. They first prove that the quasidiagonality of the reduced C*-algebra $$C_{r}^{*}(\mathbf{\Gamma})$$ implies the amenability of $$\mathbf{\Gamma}$$. For the converse direction, given an amenable unimodular discrete quantum group $${\mathbf{\Gamma}}$$, it is known by [\textit{A. Tikuisis} et al., Ann. of Math. (2) 185, No. 1, 229--284 (2017; Zbl 1367.46044)] that $$C_{r}^{*}({\mathbf{\Gamma}})$$ is quasidiagonal if it satisfies the UCT. Thus the UCT results mentioned in the previous paragraph immediately build the quasidiagonality of $$C_{r}^{*}({\mathbf{\Gamma}})$$ for those $$\mathbf{\Gamma}$$ with the cofibrant Baum-Connes property. The authors also remark that the unimodularity is necessary in this result. Reviewer: Simeng Wang (Saarbrücken) Property RD and hypercontractivity for orthogonal free quantum groups https://zbmath.org/1485.46074 2022-06-24T15:10:38.853281Z "Brannan, Michael" https://zbmath.org/authors/?q=ai:brannan.michael "Vergnioux, Roland" https://zbmath.org/authors/?q=ai:vergnioux.roland "Youn, Sang-Gyun" https://zbmath.org/authors/?q=ai:youn.sang-gyun The paper under review studies the rapid decay property (RD in short) and hypercontractivity of heat semigroups on orthogonal free quantum groups. The property RD is originally viewed as a noncommutative analogue of Sobolev type embedding $$H_{L}^{\infty}\subset C^{\infty}$$ which compares the operator norm with the $$L_{2}$$-norm under a given length function. In the quantum group setting, it was first introduced by the first author in [\textit{R. Vergnioux}, J. Operator Theory 57, no. 2, 303--324 (2007; Zbl 1120.58004)], and was adapted in [\textit{J. Bhowmick} et al., J. Noncommut. Geom. 9, no. 4, 1175--1200 (2015; Zbl 1351.46070)] with some twists to include the non-Kac case of the dual of quantum $$SU_{q}(2)$$. The first main result of this paper shows that this property RD in general does not hold for non-Kac orthogonal free quantum groups $$O_{F}^{+}$$ any more where $$F\in GL_{N}(\mathbb{C})$$ and $$F\bar{F}=\pm\mathrm{Id}_{N}$$. Instead the authors propose a weaker variant originated in [\textit{S. Vaes} and \textit{R. Vergnioux}, Duke Math. J. 140, No. 1, 35--84 (2007; Zbl 1129.46062)]. This weaker property RD turns out to be useful in the study of the hypercontractivity of heat semigroups on $$O_{F}^{+}$$. Indeed, with this new weak version, the authors establish the hypercontractivity of heat semigroups on $$O_{F}^{+}$$ for large time, and give an estimate of the optimal time of ultracontractivity, following the same scheme as [\textit{U. Franz} et al., J. Operator Theory 77, No. 1, 61--76 (2017; Zbl 1389.46088)]. Moreover, as in [\textit{E. Ricard} and \textit{Q. Xu}, Ann. Prob. 44, 867--882 (2016; Zbl 1345.46056)], using some $$L_{2}\to L_{4}$$ version of the RD inequality, they also refine the results in the Kac setting, which leads to the conjecture that the asymptotic optimal time of the $$L_2 \to L_p$$ contractivity of that heat semigroup should be approximately $$\frac{N}{2}\log (p-1)$$. Reviewer: Simeng Wang (Saarbrücken) A note on the uniqueness of Hahn-Banach extensions https://zbmath.org/1485.46075 2022-06-24T15:10:38.853281Z "Oliveira, Lina" https://zbmath.org/authors/?q=ai:oliveira.lina Summary: Let $$A$$ be a complex Banach space, and let $$A_s$$ be the symmetric part of $$A$$, i.e., the orbit of the origin under the set of all complete holomorphic vector fields on the open unit ball of $$A$$. We obtain an explicit formula for the unique norm-preserving linear extension to $$A_s$$ of a bounded linear functional defined on a norm-closed inner ideal of the normed partial Jordan *-triple $$(A,A_s)$$. An identification of the Baum-Connes and Davis-Lück assembly maps https://zbmath.org/1485.46076 2022-06-24T15:10:38.853281Z "Kranz, Julian" https://zbmath.org/authors/?q=ai:kranz.julian The paper is devoted to identifying the Baum-Connes assembly map $$\mu: K^G_*(\mathcal E_{\mathrm{Fin}}G. A) \to K_*(A \rtimes_r G)$$ for any countable discrete group $$G$$. For a group $$G$$, let $$\mathcal F = \mathrm{Fin}\; G$$ be the family of finite subgroups $$H$$ of $$G$$. An $$\mathrm{Or}(G)$$-spectrum $$\mathbf E$$ is a functor from the category of homogeneous spaces $$G/H$$ to the category of spectra. It has a natural extension to the category of $$G$$-$$CW$$-complexes and therefore defines a homology theory $$H^G_*(-,\mathbf E)$$. The Davis-Lück assembly map is $$H^G_*(\mathcal E_{\mathrm{Fin}} G, \mathbf K^G_A) \to H^G_*(pt, \mathbf K^G_A)$$. For any separable $$G$$-$$C^*$$-algebra, there exists a $$G$$-$$C^*$$-algebra $$\tilde A$$ in the localizing smallest full subcategory $$\langle \mathcal{CT}\rangle$$ of $$G$$-$$C^*$$-algebras $$\mathrm{Ind}_H^G(B)$$ induced from the subgroups $$H \subseteq G$$. It is interesting to remark that there are a natural induction isomorphism $$H_*^H(X|_H, \mathbf K^H_B) \cong H^G_*(X, \mathbf K^G_{\mathrm{Ind}^G_B})$$ and an isomorphism $$H^H_*(\mathcal E_{\mathrm{Fin}}G|_H, \mathbf K_B^H) \cong H^H_*(pt,\mathbf K^H_B)$$ for any $$G$$-$$CW$$-complex $$X$$. The author proves (Theorem~5.3) that the Meyer-Nest identification isomorphism $$D_*: K_*(\tilde A \rtimes_r G) \to K_*(A \rtimes_r G)$$ for an element $$D\in KK^G(\tilde A,A)$$ which restricted to a $$KK^H$$-equivalence for each finite subgroup $$H\subset G$$, can be extended to the homology functor isomorphisms in the diagram $\begin{tikzcd} H^G_*(\mathcal E_{\mathrm{Fin}}G, \mathbf K^G_{\tilde{A}}) \rar["\cong"] \dar["D_*"] & H^G_*(pt, \mathbf K^G_{\tilde{A}}) \dar["D_*"]\\ H^G_*(\mathcal E_{\mathrm{Fin}}G, \mathbf K^G_{\tilde{A}}) \rar["pr_*"] & H^G_*(pt, \mathbf K^G_{\tilde{A}}). \end{tikzcd}$ Reviewer: Do Ngoc Diep (Hanoi) Invertibility for some homotopy invariant functors related to Roe algebras https://zbmath.org/1485.46077 2022-06-24T15:10:38.853281Z "Makeev, G. S." https://zbmath.org/authors/?q=ai:makeev.g-s Normally a homotopy between two homomorphisms $$\phi_i :A \rightarrow B$$ for $$C^*$$-algebras $$A$$ and $$B$$ is realized by a homomorphism $$\Phi:A \rightarrow B \otimes C[0,1]$$ evaluated at the endpoints, but the author generalizes this situation to the one of being given an endofunctor $$F$$ on the category of $$C^*$$-algebras and realizing the so-called $$F$$-homotopy $$\cong_F$$ between $$\phi_0$$ and $$\phi_1$$ by endpoint evaluations of a homomorphim $$\Phi:A \rightarrow F(B \otimes C[0,1])$$. He points out that many well-known constructions like $$K$$-theory $$K_0$$, extensions semigroups Ext, and $$E$$-theories $$E_0$$ and $$E_1$$ are realized as $$F$$-homotopy classes $$[A,B]_F:= \operatorname{hom} [A,FB]/\cong_F$$ for suitably chosen $$F$$, with addition defined by taking direct sums in matrices. For $$R_X B \subseteq {\mathcal L}_B(B \otimes \ell^2(X))$$, the Roe algebra of operators of finite propagation of a space $$X$$ of bounded geometry, the author studies $$F$$-homotopy classes for the Roe functor $$F= R_X$$. By rather direct constructions he is able to give criteria on $$X$$ such that $$[A,B]_{R_X}$$ is the zero group, and when it is a group at all. He closes by giving an example of some $$X$$ which is not a group, but just a monoid. Reviewer: Bernhard Burgstaller (Florianópolis) Introduction to noncommutative topology https://zbmath.org/1485.46078 2022-06-24T15:10:38.853281Z "Skukalek, John R." https://zbmath.org/authors/?q=ai:skukalek.john-r Summary: In these notes we introduce the idea of studying topological spaces by studying algebras of continuous functions on such spaces. This leads us naturally to the study of commutative Banach algebras and the Gelfand transform, followed by commutative $$C^\ast$$-algebras and the Gelfand-Naimark theorem. This serves as the foundation for the study of noncommutative topological spaces through noncommutative $$C^\ast$$-algebras. Nonlinear $$\xi$$-Jordan $$*$$-triple derivable mappings on factor von Neumann algebras https://zbmath.org/1485.47058 2022-06-24T15:10:38.853281Z "Zhang, Fangjuan" https://zbmath.org/authors/?q=ai:zhang.fangjuan "Zhu, Xinhong" https://zbmath.org/authors/?q=ai:zhu.xinhong Summary: Let $$\mathcal{A}$$ be a factor von Neumann algebra and $$\xi$$ be a non-zero complex number. A~nonlinear map $$\phi:\mathcal{A} \to \mathcal{A}$$ has been demonstrated to satisfy $$\phi(A{\diamondsuit_\xi}B{\diamondsuit_\xi}C) = \phi(A){\diamondsuit_\xi}B{\diamondsuit_\xi}C + A{\diamondsuit_\xi} \phi(B){\diamondsuit_\xi}C + A{\diamondsuit_\xi}B{\diamondsuit_\xi}\phi(C)$$ for all $$A, B, C \in \mathcal{A}$$ if and only if $$\phi$$ is an additive $$*$$-derivation and $$\phi(\xi A) = \xi \phi(A)$$ for all $$A \in \mathcal{A}$$. Rapid decay and polynomial growth for bicrossed products https://zbmath.org/1485.81044 2022-06-24T15:10:38.853281Z "Fima, Pierre" https://zbmath.org/authors/?q=ai:fima.pierre "Wang, Hua" https://zbmath.org/authors/?q=ai:wang.hua.1|wang.hua.2|wang.hua Summary: We study the rapid decay property and polynomial growth for duals of bicrossed products coming from a matched pair of a discrete group and a compact group.