Recent zbMATH articles in MSC 46Lhttps://www.zbmath.org/atom/cc/46L2021-04-16T16:22:00+00:00WerkzeugAn operational construction of the sum of two non-commuting observables in quantum theory and related constructions.https://www.zbmath.org/1456.810102021-04-16T16:22:00+00:00"Drago, Nicolò"https://www.zbmath.org/authors/?q=ai:drago.nicolo"Mazzucchi, Sonia"https://www.zbmath.org/authors/?q=ai:mazzucchi.sonia"Moretti, Valter"https://www.zbmath.org/authors/?q=ai:moretti.valterSummary: The existence of a real linear space structure on the set of observables of a quantum system -- i.e., the requirement that the linear combination of two generally non-commuting observables \(A, B\) is an observable as well -- is a fundamental postulate of the quantum theory yet before introducing any structure of algebra. However, it is by no means clear how to choose the measuring instrument of a general observable of the form \(aA+bB\) (\(a,b\in\mathbb{R}\)) if such measuring instruments are given for the addends observables \(A\) and \(B\) when they are incompatible observables. A mathematical version of this dilemma is how to construct the spectral measure of \(f(aA+bB)\) out of the spectral measures of \(A\) and \(B\). We present such a construction with a formula which is valid for general unbounded self-adjoint operators \(A\) and \(B\), whose spectral measures may not commute, and a wide class of functions \(f: \mathbb{R}\rightarrow\mathbb{C} \). In the bounded case, we prove that the Jordan product of \(A\) and \(B\) (and suitably symmetrized polynomials of \(A\) and \(B)\) can be constructed with the same procedure out of the spectral measures of \(A\) and \(B\). The formula turns out to have an interesting operational interpretation and, in particular cases, a nice interplay with the theory of Feynman path integration and the Feynman-Kac formula.Podleś spheres for the braided quantum \(\mathrm{SU}(2)\).https://www.zbmath.org/1456.580062021-04-16T16:22:00+00:00"Sołtan, Piotr M."https://www.zbmath.org/authors/?q=ai:soltan.piotr-mikolajPodleś quantum sphere \(\mathsf{S}^2_q=\mathrm{SU}_q(2)/\mathbb{T}\), \(q\) is a real number with \(0<|q|\le 1\) is extended to the quotient of braded quantum \(\mathrm{SU}_q(2)\), \(q\) is a complex number with \(0<|q)\le 1\) [\textit{P. Kasprzak} et al., J. Noncommut. Geom. 10, No. 4, 1611--1625 (2016; Zbl 1358.81128)]. That is define (braided) quantum sphere \(\mathbb{S}^2_q\) by \(\mathbb{S}^2_q=\mathrm{SU}_q(2)/\mathbb{T}\), \(q\) a complex number \(0<|q|\le 1\).
Let \(\mathbb{S}^2_q\) be the braided quantum sphere and \(\mathsf{S}^2_q\) be the Pofleś' quantum sphere Then it is shown \(\mathbb{S}^2_q=\mathsf{S}^2_{|q|}\) (\S4. \S7, Cor. 7.3). An axiomatic definition of braided quantum sphere is also given (\S6. Remark 6.1, \S7. Def.7.2).
To define and study braded quantum \(\mathrm{SU}_q(2)\) use braided tensor product \(A\boxtimes B\). It is explained in\S2 together with related topics following [loc. cit.] and [\textit{R. Meyer} et al., Int. J. Math. 25, No. 2, Article ID 1450019, 37 p. (2014; Zbl 1315.46076)]. Braided quantum \(\mathrm{SU}_q(2)\) is explained in \S3, and shwo the algebra of functions \(C(\mathrm{SU}_q(2)\) is the uiversal \(C^\ast\)-algebras generated by two elements \(\alpha, \gamma\), together with their relations ((3.1),(3.2)) Braided quantum sphere \(\mathbb{S}^2_q\) is defined as the quotient \(\mathrm{SU}_q(2)/\mathbb{T}\) in \S4. It is shown \(C(\mathbb{S}^2_q)\) is the closed unital \(\ast\)-subalgebra of \(C(\mathrm{SU}_q(2))\) generated by \(\alpha\gamma^\ast\) and \(\gamma^\ast\gamma\) (Cor.4.3). From this corollary and [\textit{P. Podleś}, Lett. Math. Phys. 14, 193--202 (1987; Zbl 0634.46054)] \(\mathbb{S}^2_q=\mathsf{S}^2_{|q|}\) is derived.
To obtain axiomatic definition \(\mathbb{S}^2_q\), that says a compact quantum space with \(\mathbb{T}\)-action is a braided quantum sphere, if and only if there exists \(\Gamma:\mathbb{C}(\mathbb{X})\to C(\mathrm{SU}_q(2))\boxtimes_\zeta C(\mathbb{X})\);
\[(\Gamma\boxtimes_\zeta\circ\Gamma)=(\mathrm{id})\circ\Gamma=(\mathrm{id}\boxtimes_\zeta \Delta_{\mathrm{SU}_q(2)})\circ\Gamma, \]
with 4 conditions, stated in the beginning of \S6, the three dimensional irreducible representation of \(\mathrm{SU}_q(2)\) found as an irreducible subrepresentation of the tensor square of the fundamental representation is defined and studied in \S5. Then the four conditions found in \S6, from discussions in \S5 and \S4. \S7, the last Section, show that \(C^\ast\)-algebra defined by relations in \S6 indeed carry and action of the braided \(\mathrm{SU}_q(2)\) and that they coincide with Podleś sphere (Def.7,2. Cor. 7.3).
Some hard calculus in the study of the tensor product of the fundamental representation are given in Appendix.
Reviewer: Akira Asada (Takarazuka)Bounded derivations on uniform Roe algebras.https://www.zbmath.org/1456.460572021-04-16T16:22:00+00:00"Lorentz, Matthew"https://www.zbmath.org/authors/?q=ai:lorentz.matthew"Willett, Rufus"https://www.zbmath.org/authors/?q=ai:willett.rufusThis paper studies bounded derivations of uniform Roe algebras, operator algebras capable of detecting phenomena in coarse geometry. These algebras (and their nonuniform analogues) were introduced by \textit{J. Roe} [Coarse cohomology and index theory on complete Riemannian manifolds. Providence, RI: American Mathematical Society (AMS) (1993; Zbl 0780.58043)] for their connections with (higher) index theory, specifically for their applications to elliptic operators on noncompact manifolds [\textit{J. Roe}, Index theory, coarse geometry, and topology of manifolds. Providence, RI: AMS, American Mathematical Society (1996; Zbl 0853.58003)]; later the study of these algebras was boosted due to its intrinsic relation with the coarse Baum-Connes conjecture and, consequently, with the Novikov conjecture [\textit{G.-L. Yu}, Invent. Math. 139, No. 1, 201--240 (2000; Zbl 0956.19004)]. Recently they have also been used as a framework in mathematical physics to study the classification of topological phases and the topology of quantum systems (see, e.g., [\textit{E. E. Ewert} and \textit{R. Meyer}, Commun. Math. Phys. 366, No. 3, 1069--1098 (2019; Zbl 07041901)]).
This paper focuses on a purely algebraic aspect of uniform Roe algebras: the study of their derivations. A derivation on a \(C^*\)-algebra \(A\) is a linear map \(\delta: A\to A\) with the property that
\[
\delta(ab)=a\delta(b)+\delta(a)b
\]
for all \(a,b\in A\). If \(A\) is unital and \(d\in A\), the map \(a\mapsto da-ad\) is a derivation; these derivations are called inner. The question of which \(C^*\)-algebras have only inner derivations was studied because of its connections with mathematical physics and one-parameters groups. It was shown (see [\textit{C. A. Akemann} and \textit{G. K. Pedersen}, Am. J. Math. 101, 1047--1061 (1979; Zbl 0432.46059); \textit{G. A. Elliott}, Ann. Math. (2) 106, 121--143 (1977; Zbl 0365.46051)]) that the only separable \(C^*\)-algebras having only inner bounded derivations are those that can be written as a direct sum of continuous trace and simple blocks. On the other side of the spectrum, von Neumann algebras only have inner bounded derivations [\textit{S. Sakai}, Ann. Math. (2) 83, 273--279 (1966; Zbl 0139.30601)].
Uniform Roe algebras of uniformly locally finite spaces are nonseparable, and far from being simple or abelian. The main result of this paper shows that these objects (which, even though they are not von Neumann algebras, can be written as a countable union of weakly closed Banach subspaces) retain some sort of rigidity when it comes to derivations. In particular, they only have inner bounded derivations. This theorem describes a completely new class of objects which only have inner derivations.
The proof has two main ingredients: a basic form of a ``reduction of cocycles'' argument of \textit{A. M. Sinclair} and \textit{R. R. Smith} [Contemp. Math. 365, 383--400 (2004; Zbl 1080.46039)] used in studying Hochschild cohomology of von Neumann algebras, and recent Ramsey-theoretic ideas [\textit{B. M. Braga} and \textit{I. Farah}, Trans. Am. Math. Soc. 374, No. 2, 1007--1040 (2021; Zbl 07291890)] used in the study of uniform Roe algebras.
Reviewer: Alessandro Vignati (Toronto)Certain \(\ast\)-homomorphisms acting on unital \(C^\ast\)-probability spaces and semicircular elements induced by \(p\)-adic number fields over primes \(p\).https://www.zbmath.org/1456.460532021-04-16T16:22:00+00:00"Cho, Ilwoo"https://www.zbmath.org/authors/?q=ai:cho.ilwooSummary: In this paper, we study the Banach \(*\)-probability space
\((A\otimes_{\mathbb{C}}\mathbb{LS}, \tau_A^0)\) generated by a fixed unital \(C^*\)-probability space \((A, \varphi_A)\), and the semicircular elements \(\Theta_{p,j}\) induced by \(p\)-adic number fields \(\mathbb{Q}_p\), for all \(p \in \mathcal{P}\), \(j\in\mathbb{Z}\), where \(\mathcal{P}\) is the set of all primes, and \(\mathbb{Z}\) is the set of all integers. In particular, from the order-preserving shifts \(g\times h_\pm\) on \(\mathcal{P} \times \mathbb{Z}\), and \(*\)-homomorphisms \(\theta\) on \(A\), we define the corresponding \(*\)-homomorphisms \(\sigma_{(\pm ,1)}^{1:\theta}\) on \(A\otimes_{\mathbb{C}} \mathbb{LS}\), and consider free-distributional data affected by them.Classification of certain inductive limit actions of compact groups on AF algebras.https://www.zbmath.org/1456.190042021-04-16T16:22:00+00:00"Wang, Qingyun"https://www.zbmath.org/authors/?q=ai:wang.qingyunThe paper is devoted to group actions on AF-algebras and classification upto unitary conjugacy. Let \(G\) be a compact group acting \(\alpha=\varinjlim \alpha_n: G \curvearrowright A, \alpha_n: G \curvearrowright A_n\) stably on each entry \(A_n\) of the inductive limit \(A = \varinjlim A_n\). Following the Elliot's classification program, the classifying objects \(\mathrm{Ell}(\alpha):=\mathrm{Ell}(G,A,\alpha)\) are consisting of ordered \(K\)-groups with positive cone \((K_0^\lambda(\alpha), K_0^\lambda(\alpha)_+)\) for any 2-cocycles \(\lambda\) on \(G\). To each 2-cocycle \(\lambda\) associate the Grothendieck ring \(R^\lambda(G)\) of the semigroups \(V^\lambda(G)\) of equivalent classes of \(\lambda\)-representations of \(G\). This groups make an action on the Elliot objects
$(K_0^{\lambda_2}(G),K_0^{\lambda_2}(G)_+) \overset{R^{\lambda_1}(G) \curvearrowright}{\longrightarrow}(K_0^{\lambda_1\lambda_2}(G), K_0^{\lambda_1\lambda_2}(G)_+)$
commuting with morphisms of Elliot objects. The main result (Theorem 4.2) of the paper states that any two locally spectrally trivial actions \(\alpha: G \curvearrowright A\) and \(\beta: G \curvearrowright B\) of a compact group \(G\) on both unital (or both non-unital) together AF-algebras \(A\) and \(B\) are conjugate if and only if the corresponding Elliot objects are isomorphic \(\mathrm{Ell}(G,A,\alpha) \cong \mathrm{Ell}(G,A,\beta)\). The proof of this theorem is similar to the one of classification of AF-algebras.
Reviewer: Do Ngoc Diep (Hanoi)Petz reconstruction in random tensor networks.https://www.zbmath.org/1456.813422021-04-16T16:22:00+00:00"Jia, Hewei Frederic"https://www.zbmath.org/authors/?q=ai:jia.hewei-frederic"Rangamani, Mukund"https://www.zbmath.org/authors/?q=ai:rangamani.mukundSummary: We illustrate the ideas of bulk reconstruction in the context of random tensor network toy models of holography. Specifically, we demonstrate how the Petz [\textit{D. Petz}, Commun. Math. Phys. 105, 123--131 (1986; Zbl 0597.46067)] reconstruction map works to obtain bulk operators from the boundary data by exploiting the replica trick. We also take the opportunity to comment on the differences between coarse-graining and random projections.Ergodic actions of compact quantum groups from solutions of the conjugate equations.https://www.zbmath.org/1456.370112021-04-16T16:22:00+00:00"Pinzari, Claudia"https://www.zbmath.org/authors/?q=ai:pinzari.claudia"Roberts, John E."https://www.zbmath.org/authors/?q=ai:roberts.john-eliasSummary: We use a tensor \(C^{\ast}\)-category with conjugates and two quasitensor functors into the category of Hilbert spaces to define a \({}^{\ast}\)-algebra depending functorially on this data. If one of them is tensorial, we can complete in the maximal \(C^{\ast}\)-norm. A particular case of this construction allows us to begin with solutions of the conjugate equations and associate ergodic actions of quantum groups on the \(C^{\ast}\)-algebra in question. The quantum groups involved are \(A_{u}(Q)\) and \(B_{u}(Q)\).Convolution type \(C^*\)-algebras.https://www.zbmath.org/1456.460442021-04-16T16:22:00+00:00"Nourouzi, Kourosh"https://www.zbmath.org/authors/?q=ai:nourouzi.kourosh"Reza, Ali"https://www.zbmath.org/authors/?q=ai:reza.aliSummary: In this paper, by using the notion of convolution types we introduce symmetric and non-symmetric convolution type \(C^*\)-algebras. It is shown that any (exact) convolution type induces a (an exact) functor on the category of \(C^*\)-algebras. In particular, any group induces a convolution type and a functor on the category of \(C^*\)-algebras. It is also shown that discrete crossed product of \(C^*\)-algebras and discrete inverse semigroup \(C^*\)-algebras can be considered as convolution type \(C^*\)-algebras.Square functions for noncommutative differentially subordinate martingales.https://www.zbmath.org/1456.460522021-04-16T16:22:00+00:00"Jiao, Yong"https://www.zbmath.org/authors/?q=ai:jiao.yong"Randrianantoanina, Narcisse"https://www.zbmath.org/authors/?q=ai:randrianantoanina.narcisse"Wu, Lian"https://www.zbmath.org/authors/?q=ai:wu.lian"Zhou, Dejian"https://www.zbmath.org/authors/?q=ai:zhou.dejianSummary: We prove inequalities involving noncommutative differentially subordinate martingales. More precisely, we prove that if \(x\) is a self-adjoint noncommutative martingale and \(y\) is weakly differentially subordinate to \(x\) then \(y\) admits a decomposition \(dy = a + b + c\) (resp. \( dy = z + w)\) where \(a, b\), and \(c\) are adapted sequences (resp., \(z\) and \(w\) are martingale difference sequences) such that:
\[ \| (a_n)_{n \geq 1} \|_{L_{1, \infty} (\mathcal{M} \overline{\otimes} \ell_{\infty})} + \left\| \left(\sum_{n \geq 1} \mathcal{E}_{n-1} |b_n |^2 \right)^{1/2} \right\|_{1, \infty}
+ \left\| \left(\sum_{n \geq 1} \mathcal{E}_{n-1} |c_n^* |^2 \right)^{1/2} \right\|_{1, \infty} \leq C \| x \|_1
\]
\[
\text{(resp., } \left\| \left(\sum_{n \geq 1} |z_n|^2 \right)^{{1}/{2}}\right\|_{1, \infty}+ \left\| \left(\sum_{n \geq 1} |w_n^*|^2 \right)^{1/2} \right\|_{1, \infty} \leq C \| x \|_1).
\]
We also prove strong-type \((p,p)\) versions of the above weak-type results for \(1 < p < 2\). In order to provide more insights into the interactions between noncommutative differential subordinations and martingale Hardy spaces when \(1 \leq p < 2\), we also provide several martingale inequalities with sharp constants which are new and of independent interest. As a byproduct of our approach, we obtain new and constructive proofs of both the noncommutative Burkholder-Gundy inequalities and the noncommutative Burkholder/Rosenthal inequalities for \(1 < p < 2\) with the optimal order of the constants when \(p \to 1\).Fusion frames for operators in Hilbert \(C^\ast\)-modules.https://www.zbmath.org/1456.420402021-04-16T16:22:00+00:00"Khayyami, M."https://www.zbmath.org/authors/?q=ai:khayyami.mahdiyeh"Nazari, A."https://www.zbmath.org/authors/?q=ai:nazari.akbarSummary: In this paper we introduce \(K\)-fusion frames on a Hilbert \(C^\ast\)-module \(H\), where \(K\) is an adjointable operator on \(H\). We obtain several characterizations of \(K\)-fusion frames. In addition, we extend the concept of duality to \(K\)-fusion frames and study some of its properties.Corrigendum to: ``Completely bounded norms of right module maps''.https://www.zbmath.org/1456.460472021-04-16T16:22:00+00:00"Levene, Rupert H."https://www.zbmath.org/authors/?q=ai:levene.rupert-h"Timoney, Richard M."https://www.zbmath.org/authors/?q=ai:timoney.richard-mThis correction refers to [the authors, ibid. 436, No. 5, 1406--1424 (2012; Zbl 1244.46026)].On fundamental groups of tensor product \(\text{II}_1\) factors.https://www.zbmath.org/1456.460492021-04-16T16:22:00+00:00"Isono, Yusuke"https://www.zbmath.org/authors/?q=ai:isono.yusukeSummary: Let \(M\) be a \(\text{II}_1\) factor and let \(\mathcal{F}(M)\) denote the fundamental group of \(M\). In this article, we study the following property of \(M\): for any \(\text{II}_1\) factor \(B\), we have \(\mathcal{F}(M\overline\otimes B)=\mathcal{F}(M)\mathcal{F}(B)\). We prove that for any subgroup \(G\leqslant\mathbb{R}_+^\ast\) which is realized as a fundamental group of a \(\text{II}_1\) factor, there exists a \(\text{II}_1\) factor \(M\) which satisfies this property and whose fundamental group is \(G\). Using this, we deduce that if \(G,H\leqslant\mathbb{R}_+^\ast\) are realized as fundamental groups of \(\text{II}_1\) factors, then so are groups \(G\cdot H\) and \(G\cap H\).Extending representations of Banach algebras to their biduals.https://www.zbmath.org/1456.460422021-04-16T16:22:00+00:00"Gardella, Eusebio"https://www.zbmath.org/authors/?q=ai:gardella.eusebio"Thiel, Hannes"https://www.zbmath.org/authors/?q=ai:thiel.hannesLet \(\varphi\) be a representation of a Banach algebra \(A\) on a Banach space \(X\). The essential space \(X_\varphi\) of \(\varphi\) is the closed linear span of \{\(\varphi(a)x$,\, $a \in A$,\, $x \in X\)\}. Firstly, the authors give conditions so that there is an extension \(\widetilde{\varphi}\) of \(\varphi\) on the bidual (Banach) algebra of \(A\), under (the left or yet the right) Arens product; in that case, the essential spaces of \(\varphi\) and \(\widetilde{\varphi}\) agree. Based on this result, the authors deal with the complementarity of the essential spaces (with respect to representations of Banach algebras). A positive result is given when \(A\) has a bounded left approximate identity and every operator \(A \rightarrow X\) is weakly compact.
Further, the authors connect the existence of representations of certain Banach algebras to that of the respective (left) multiplier algebras. In particular, they consider the case when \(X\) belongs to a class of reflexive Banach spaces, closed under complementation (of its subspaces) and \(A\) is a certain representable Banach algebra with a contractive left approximate identity. Actually, \(A\) accepts a nondegenerate isometric representation on some space like \(X\). This implies that the left multiplier algebra of \(A\) has a unital,
isometric representation on a space \(X\), as before. This is also true for any \(C^\ast\)-algebra being isometrically representable on an \(L^p\)-space, \(p \in [1, \infty)\). Based, amongst others, on the latter result, the authors prove that, in the context of \(C^\ast\)-algebras \(A\), the (ring) commutativity characterizes such an \(A\) as isometrically represented on an \(L^p\)-space, \(p \in [1, \infty) \backslash \{2\}\).
Moreover, for a locally compact group \(G\), the completion of \(L^1(G)\) for nondegenerate representations
on \(L^p\)-spaces is called the universal group \(L^p\)-operator algebra of \(G\). Actually, the latter is universal for all contractive representations of \(L^1(G)\) on \(L^p\)-spaces.
Applications of the present paper are given by the authors in [Trans. Am. Math. Soc. 371, 2207--2236 (2019; Zbl 06999077)].
Reviewer: Marina Haralampidou (Athína)A tool kit for groupoid $C^*$-algebras.https://www.zbmath.org/1456.460022021-04-16T16:22:00+00:00"Williams, Dana P."https://www.zbmath.org/authors/?q=ai:williams.dana-pGroupoids are small categories in which every morphism is an isomorphism.
They appear as early as in the work of \textit{H. Brandt} [Mat. Ann. 96, 360--366 (1926; JFM 52.0110.09)] on generalising a composition of binary quadratic forms (due to Gauss) to quaternary quadratic forms.
Since then, groupoids have been used in a wide variety of areas of mathematics including ergodic theory, functional analysis, homotopy theory, algebraic geometry, differential geometry, differential topology, and group theory.
The book under review focuses on the interactions with operator algebras that span more than four decades of research.
In the late 1970s, Renault initiated the idea of associating a \(C^*\)-algebra to a locally compact groupoid in analogy to what is done for groups.
Since then, a large amount of research has emerged concerning the structure and properties of such a \(C^*\)-algebra.
Starting with the foundational work of \textit{J. Renault} [A groupoid approach to \(C^*\)-algebras. Berlin: Springer (1980; Zbl 0433.46049)] and \textit{A. Kumjian} [Can. J. Math. 38, 969--1008 (1986; Zbl 0627.46071)], the current book aims to present the core theory of groupoid \(C^*\)-algebras by taking advantage of recent developments covered by the book of \textit{A. L. T. Paterson} [Groupoids, inverse semigroups, and their operator algebras. Boston, MA: Birkhäuser (1999; Zbl 0913.22001)], \textit{P. Muhly}'s unpublished CBMS lecture notes [``Coordinates in operator algebra'', CBMS Conference Lecture Notes (Texas Christian University 1990), unfinished manuscript (1999), \url{https://operatoralgebras.org/resources-resources/Groupoids-Book-Muhly.pdf}], and the monograph by \textit{C. Anantharaman-Delaroche} and \textit{J. Renault} [Amenable groupoids. Genève: L'Enseignement Mathématique; Université de Genève (2000; Zbl 0960.43003)].
The book covers four topics for second countable, locally compact Hausdorff groupoids that are at the epicentre in this endeavour:
\begin{itemize}
\item[(a)] the disintegration theorem;
\item[(b)] the equivalence theorem;
\item[(c)] three notions of amenability (topological amenability, measurewise amenability and metric amenability) and their relations;
\item[(d)] simplicity (in the sense of Effros-Hahn-type results).
\end{itemize}
The author uses current developments to provide simplified proofs with a clear exposition. In the process, all required tools are carefully laid down. Some technical parts of proofs are included as exercises with solutions at the end that allow the reader to dig into the fine details of the material.
The book is self-contained for a reader with some knowledge in functional analysis, and while experience in working with group \(C^*\)-algebras is -- always -- helpful, it is not much required. The material is accessible to graduate students that wish to undertake work in this direction.
Reviewer: Evgenios Kakariadis (Newcastle upon Tyne)Hyperreflexivity constants of the bounded \(n\)-cocycle spaces of group algebras and \(C^*\)-algebras.https://www.zbmath.org/1456.460452021-04-16T16:22:00+00:00"Samei, Ebrahim"https://www.zbmath.org/authors/?q=ai:samei.ebrahim"Farsani, Jafar Soltani"https://www.zbmath.org/authors/?q=ai:farsani.jafar-soltaniSummary: We introduce the concept of strong property \((\mathbb{B})\) with a constant for Banach algebras and, by applying a certain analysis on the Fourier algebra of the unit circle, we show that all \(C^*\)-algebras and group algebras have the strong property \((\mathbb{B})\) with a constant given by \(288 \pi (1 + \sqrt{2})\). We then use this result to find a concrete upper bound for the hyperreflexivity constant of \(\mathcal{Z}^n (A, X)\), the space of bounded \(n\)-cocycles from \(A\) into \(X\), where \(A\) is a \(C^*\)-algebra or the group algebra of a group with an open subgroup of polynomial growth and \(X\) is a Banach \(A\)-bimodule for which \(\mathcal{H}^{n+1} (A, X)\) is a Banach space. As another application, we show that for a locally compact amenable group \(G\) and \(1 < p < \infty\), the space \(CV_P(G)\) of convolution operators on \(L^p(G)\) is hyperreflexive with a constant given by \(384 \pi^2 (1 + \sqrt{2})\). This is the generalization of a well-known result of \textit{E. Christensen} [Math. Scand. 50, 111--122 (1982; Zbl 0503.47032)] for \(p = 2\).Equivariant dimensions of graph \(C^\ast\)-algebras.https://www.zbmath.org/1456.190032021-04-16T16:22:00+00:00"Chirvasitu, Alexandru"https://www.zbmath.org/authors/?q=ai:chirvasitu.alexandru"Passer, Benjamin"https://www.zbmath.org/authors/?q=ai:passer.benjamin-w"Tobolski, Mariusz"https://www.zbmath.org/authors/?q=ai:tobolski.mariuszLet \(E = (E^0, E^1, r, s)\) be a graph with countable vertex set \(E^0\), countable edge set \(E^1\), the rival and source maps \(r,s : E^1 \to E^0\) and the adjacency matrix \(A_E = (A_{vw}), A_{vw} = \#\{ \mbox{edges with source \textit{v} and rival \textit{w}}\}\). One associates to \(E\) a g\textit{raph \(C^\ast\)-algebra} \(C^*(E)\) generated by the mutually orthogonal projections \(P_v, v\in E^0\) corresponding to vertices and the mutually orthogonal partial isometries \(S_e, e\in E^1\) corresponding to edges satisfying the conditions: for each \(e\in E^1\),
\(S_e^*S_e= P_{r(e)}\), \(S_eS_e^* = P_{s(e)}\), and for each \(v\in E^0\), \(P_v = \sum_{e\in s^{-1}(v)} S_eS^*_e\). The gauge action \(\mathbb S^1 \curvearrowright C^*(E)\) is defined by \(S_e \mapsto \lambda E_e\) and \(P_v \mapsto P_v, \forall \lambda\in \mathbb S^1\). The restriction of the \textit{gauge action} to subgroup \(\mathbb Z/k \hookrightarrow \mathbb S^1\). For a subgroup \(G\) acting on a unital \(C^\ast\)-algebra \(A\), the \textit{local-triviality dimension} \(\dim_{LT}^G(A)\) is the smallest \(n\) for which there exist \(G\)-equivalent *-homomorphism \(\rho_0, \dots, \rho_n: C_0((0,1]) \otimes C(G) \to A\) such that \(\sum_{i=0}^n \rho_i(t\otimes 1) = 1\). The weak (resp., strong) local-triviality dimension \(\dim_{WLT}^G(A)\)(resp., \(\dim_{SLT}^G(A)\)) is the smallest \(n\) for which there exist \(G\)-equivalent *-homomorphism \(\rho_0, \dots, \rho_n: C_0((0,1]) \otimes C(G) \to A\) such that \(\sum_{i=0}^n \rho_i(t\otimes 1)\) is invertible (resp. there is a unital *-homomorphism \(C(E_nG) \to A\), \(E_nG := E_{n-a}G*G, E_0G:= G)\). It is clear that \(\dim_{WLT}^G(A) \leq \dim_{LT}^G(A) \leq \dim_{SLT}^G(A) \).
For \(C^\ast\)-algebras of finite acyclic graphs and finite cycles, as the main result, the authors
\textit{characterize the finiteness of these dimensions} (Theorems 3.4, 4.1, 4.4), and then
study the gauge actions on various examples of graph \(C^\ast\)-algebras, including Cuntz algebras (\S5.1), the Toeplitz algebra (\S5.2), and the antipodal actions on quantum spheres (\S5.4).
Reviewer: Do Ngoc Diep (Hanoi)Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem.https://www.zbmath.org/1456.600132021-04-16T16:22:00+00:00"Belinschi, Serban T."https://www.zbmath.org/authors/?q=ai:belinschi.serban-teodor"Mai, Tobias"https://www.zbmath.org/authors/?q=ai:mai.tobias"Speicher, Roland"https://www.zbmath.org/authors/?q=ai:speicher.rolandSummary: We develop an analytic theory of operator-valued additive free convolution in terms of subordination functions. In contrast to earlier investigations our functions are not just given by power series expansions, but are defined as Fréchet analytic functions in all of the operator upper half plane. Furthermore, we do not have to assume that our state is tracial. Combining this new analytic theory of operator-valued free convolution with Anderson's selfadjoint version of the linearization trick we are able to provide a solution to the following general random matrix problem: Let \(X_1^{(N)},\dots,X_n^{(N)}\) be selfadjoint \(N\times N\) random matrices which are, for \(N\), asymptotically free. Consider a selfadjoint polynomial \(p\) in \(n\) non-commuting variables and let \(P^{(N)}\) be the element \(P^{(N)}=p(X_1^{(N)},\dots,X_n^{(N)})\). How can we calculate the asymptotic eigenvalue distribution of \(P^{(N)}\) out of the asymptotic eigenvalue distributions of \(X_1^{(N)},\dots,X_n^{(N)}\)?About the foundation of the Kubo generalized cumulants theory: a revisited and corrected approach.https://www.zbmath.org/1456.600472021-04-16T16:22:00+00:00"Bianucci, Marco"https://www.zbmath.org/authors/?q=ai:bianucci.marco"Bologna, Mauro"https://www.zbmath.org/authors/?q=ai:bologna.mauroThe joint numerical radius on \(C^*\)-algebras.https://www.zbmath.org/1456.460432021-04-16T16:22:00+00:00"Mabrouk, Mohamed"https://www.zbmath.org/authors/?q=ai:mabrouk.mohamedSummary: Let \(\mathfrak{A}\) be unital \(C^*\)-algebra with unit \(e\) and positive cone \(\mathfrak{A}^+\) such that every irreducible representation is infinite dimensional. For every \(\mathbf{a} =(a_1,\dots,a_n)\in\mathfrak{A}^n\), the joint numerical radius of \(\mathbf{a}\) is denoted by \(\mathbf{v}(\mathbf{a})\). It is shown that an element \(\mathbf{a}\in \mathfrak{A}^n\) satisfies \(\sum_{j=1}^n|f(a_j)|^2=1\) for every pure state \(f\) of \(\mathfrak{A}\) if and only if each \(a_j\) is in the center of \(\mathfrak{A}\) and \( \sum_{j=1}^na_j a_j^*=e\). Furthermore, we characterize elements \(\mathbf{a}_1, \dots, \mathbf{a}_n \in \mathfrak{A}^n\) such that for any \(\mathbf{x}\in (\mathfrak{A}^+)^n\) there exists \(\alpha =(\alpha_1, \dots, \alpha_n)\in \mathbb{R}^n\) such that \( \sum_{j=1}^{j=n}\alpha_j^2=1\) and \(\mathbf{v}(\sum_{j=1}^{j=n}\alpha_j\mathbf{a}_j+ \mathbf{x}) = 1+\mathbf{v}(\mathbf{x})\).
For the entire collection see [Zbl 1444.15003].Covariant representations for possibly singular actions on \(C^*\)-algebras.https://www.zbmath.org/1456.460552021-04-16T16:22:00+00:00"Beltiţă, Daniel"https://www.zbmath.org/authors/?q=ai:beltita.daniel"Grundling, Hendrik"https://www.zbmath.org/authors/?q=ai:grundling.hendrik-b|grundling.hendrik-b-g-s"Neeb, Karl-Hermann"https://www.zbmath.org/authors/?q=ai:neeb.karl-hermannSummary: Singular actions on \(C^*\)-algebras are automorphic group actions on \(C^*\)-algebras, where the group is not locally compact, or the action is not strongly continuous. We study the covariant representation theory of actions which may be singular. In the usual case of strongly continuous actions of locally compact groups on \(C^*\)-algebras, this is done via crossed products, but this approach is not available for singular \(C^*\)-actions. We explored extension of crossed products to singular actions in a previous paper [\textit{H. Grundling} and \textit{K.-H. Neeb}, J. Funct. Anal. 266, No. 8, 5199--5269 (2014; Zbl 1303.46059)]. The literature regarding covariant representations for possibly singular actions is already large and scattered, and in need of some consolidation. We collect in this survey a range of results in this field, mostly known. We improve some proofs and elucidate some interconnections. These include existence theorems by Borchers and Halpern, Arveson spectra, the Borchers-Arveson theorem, standard representations and Stinespring dilations as well as ground states, KMS states and ergodic states and the spatial structure of their GNS representations.Bott periodicity and almost commuting matrices.https://www.zbmath.org/1456.460602021-04-16T16:22:00+00:00"Willett, Rufus"https://www.zbmath.org/authors/?q=ai:willett.rufusSummary: We give a proof of the Bott periodicity theorem for topological \(K\)-theory of \(C^*\)-algebras based on Loring's treatment of Voiculescu's almost commuting matrices and Atiyah's rotation trick. We also explain how this relates to the Dirac operator on the circle; this uses Yu's localization algebra and an associated explicit formula for the pairing between the first \(K\)-homology and first \(K\)-theory groups of a (separable) \(C^*\)-algebra.
For the entire collection see [Zbl 1441.19002].Noncommutative counterparts of celebrated conjectures.https://www.zbmath.org/1456.460632021-04-16T16:22:00+00:00"Tabuada, Gonçalo"https://www.zbmath.org/authors/?q=ai:tabuada.goncaloSummary: In this survey, written for the proceedings of the \textit{\(K\)-theory conference}, Buenos Aires and La Plata, Argentina (satellite event of the ICM 2018), we give a rigorous overview of the noncommutative counterparts of some celebrated conjectures of Grothendieck, Voevodsky, Beilinson, Weil, Tate, Parshin, Kimura, Schur, and others.
For the entire collection see [Zbl 1441.19002].Unit-free contractive projection theorems for \(C^\ast\)-, \(JB^\ast\)-, and \(JB\)-algebras.https://www.zbmath.org/1456.460592021-04-16T16:22:00+00:00"Cabrera García, Miguel"https://www.zbmath.org/authors/?q=ai:cabrera-garcia.miguel"Rodríguez Palacios, Ángel"https://www.zbmath.org/authors/?q=ai:rodriguez-palacios.angelFor a (possibly non-associative) complete normed algebra \(A\), the range of any contractive linear projection \(\pi :A\to A\) becomes a complete normed algebra under the restriction of the norm of \(A\) and the product \(\odot ^\pi\) defined by \(x\odot ^\pi y:=\pi (xy)\) for all \(x,y \in \pi (A)\).
The contractive projection problem for a class \(\mathcal{C}\) of complete normed algebras reads as follows:
Given an arbitrary member \(A\) of \(\mathcal{C}\), and any contractive linear projection \(\pi:A\to A\) (sometimes subjected to some additional natural requirements), does \((\pi(A),\odot^\pi)\) lie in \(\mathcal{C}\)?
It is well known that the contractive projection problem has a negative answer when \(\mathcal{C}\) is the class of all unital \(C^*\)-algebras, and the extra requirement on \(\pi \) is that \(\pi (\mathbf{1})=\mathbf{1}\). Nevertheless, relevant affirmative answers to the contractive projection problem have been proved in the literature for \(C^*\)-algebras and for different classes generalizing the \(C^*\)-algebras (\(JC\)- and \(JB\)-algebras, alternative \(C^*\)-algebras, \(JC^*\)- and \(JB^*\)-algebras, non-commutative \(JB^*\)-algebras, Arazy algebras, and associative or Jordan operator algebras).
In the paper under review, it is proved that the contractive projection problem has an affirmative answer in the following cases:
\begin{itemize}
\item[-] \(\mathcal{C}\) is the class of all commutative \(C^*\)-algebras or that of all non-com\-mutative \(JB^*\)-algebras, and positivity of \(\pi \) is the unique extra requirement on \(\pi \).
\item[-] \(\mathcal{C}\) is the class of all \(JB\)-algebras or that of all \(JC\)-algebras, and positivity of \(\pi \) is the unique extra requirement on \(\pi \).
\item[-] \(\mathcal{C}\) is the class of all \(C^*\)-algebras or that of all alternative \(C^*\)-algebras, and \(\pi (\pi (a)^*\pi (a))\leq \pi (a^*a)\) for all \(a\in A\in \mathcal{C}\) and positivity of \(\pi \) are the extra requirements on \(\pi \).
\item[-] \(\mathcal{C}\) is the class of all non-commutative \(JB^*\)-algebras, that of all \(JC^*\)-algebras, or that of all commutative \(C^*\)-algebras, and the validity of the equality \(\pi (\pi (a) \pi (b)+ \pi (b) \pi (a))=\pi (a\pi (b)+\pi (b)a)\) for all \(a,b\in A\in \mathcal{C}\)
is the unique extra requirement on \(\pi \).
\item[-] \(\mathcal{C}\) is the class of all \(C^*\)-algebras or that of all alternative \(C^*\)-algebras, and the validity of the equalities \(\pi (\pi (a)\pi (b))=\pi (a\pi (b))=\pi (\pi (a)b)\) for all \(a,b\in A\in \mathcal{C}\) is the unique extra requirement on \(\pi \).
\end{itemize}
Moreover, the paper contains a full description of positive bi-contractive linear projections on non-commutative \(JB^*\)-algebras, as well as a structure theorem for bi-contractive linear projections (without any extra requirement) on non-commutative \(JBW^*\)-algebras. In addition, bi-contractive linear projections on \(C^*\)-algebras are studied in detail.
Reviewer: Antonio Fernández López (Málaga)Finite-dimensional approximations for Nica-Pimsner algebras.https://www.zbmath.org/1456.460482021-04-16T16:22:00+00:00"Kakariadis, Evgenios T. A."https://www.zbmath.org/authors/?q=ai:kakariadis.evgenios-t-aThe author presents necessary and sufficient conditions for nuclearity of Cuntz-Nica-Pimsner algebras for a variety of quasi-lattice ordered groups. Firstly, he proves necessary and sufficient conditions for nuclearity for \(\mathbb{Z}_+^N\)-product systems. Secondly, he abstracts his methods to accommodate more general quasi-lattices that attain a \(\mathbb{Z}_+^N\)-controlled map in the sense of [\textit{J. Crisp} and \textit{M. Laca}, J. Funct. Anal. 242, No. 1, 127--156 (2007; Zbl 1112.46051)]. This class includes quasi-lattices such as the Baumslag-Solitar group for \(n=m>0\) and the right-angled Artin groups. His arguments tackle Nica-Pimsner algebras that admit a faithful conditional expectation on a small fixed point algebra and a faithful copy of the coefficient algebra. Splitting the fixed point algebra, he shows that exactness of the underlying \(C^\ast\)-algebra is equivalent to exactness of the Toeplitz-Nica-Pimnser algebra.
Reviewer: Mohammad Sal Moslehian (Mashhad)On graph products of multipliers and the Haagerup property for \(C^{\ast}\)-dynamical systems.https://www.zbmath.org/1456.460542021-04-16T16:22:00+00:00"Atkinson, Scott"https://www.zbmath.org/authors/?q=ai:atkinson.scott-eThe following paragraphs are essentially taken from the author's abstract and introduction.
The Haagerup property is an important approximation property for groups and for self-adjoint operator algebras. Since its appearance in Haagerup's seminal article, this property has been the subject of intense study. In 2012, \textit{Z. Dong} and \textit{Z.-J. Ruan} [Integral Equations Oper. Theory 73, No. 3, 431--454 (2012; Zbl 1263.46043)] introduced the Haagerup property for the action of a discrete group \(G\) on a unital \(C^*\)-algebra \(A\).
The author considers the notion of the graph product of actions of discrete groups \(\{G_v\}\) on a \(C^*\)-algebra \(A\) and shows that, under suitable commutativity conditions, the graph product action \(\bigstar_\Gamma \alpha_v:\bigstar_\Gamma G_v\curvearrowright A\) has the Haagerup property if each action \(\alpha_v: G_v\curvearrowright A\) possesses the Haagerup property. This generalizes the known results on graph products of groups with the Haagerup property. To accomplish this, the author introduces the graph product of multipliers associated to the actions and shows that the graph product of positive-definite multipliers is positive definite. These results have impacts on left-transformation groupoids and give an alternative proof of a known result for coarse embeddability. The author also records a cohomological characterization of the Haagerup property for group actions.
Reviewer: Qing Meng (Qufu)Structure of block quantum dynamical semigroups and their product systems.https://www.zbmath.org/1456.460582021-04-16T16:22:00+00:00"Rajarama Bhat, B. V."https://www.zbmath.org/authors/?q=ai:bhat.b-v-rajarama"Vijaya Kumar, U."https://www.zbmath.org/authors/?q=ai:vijaya-kumar.uA product expansion for Toeplitz operators on the Fock space.https://www.zbmath.org/1456.470072021-04-16T16:22:00+00:00"Hagger, Raffael"https://www.zbmath.org/authors/?q=ai:hagger.raffaelSummary: We study the asymptotic expansion of the product of two Toeplitz operators on the Fock space. In comparison to earlier results, we require significantly fewer derivatives and get the expansion to arbitrary order. This, in particular, improves a result of Borthwick related to Toeplitz quantization [\textit{D. Borthwick}, Contemp. Math. 214, 23--37 (1998; Zbl 0903.58013)]. In addition, we derive an intertwining identity between the Berezin star product and the sharp product.Permanence of weak comparison for large subalgebras.https://www.zbmath.org/1456.460462021-04-16T16:22:00+00:00"Zhao, Xia"https://www.zbmath.org/authors/?q=ai:zhao.xia"Fang, Xiaochun"https://www.zbmath.org/authors/?q=ai:fang.xiaochun"Fan, Qingzhai"https://www.zbmath.org/authors/?q=ai:fan.qingzhaiSummary: Let \(A\) be an infinite dimensional simple unital stably finite \(C^*\)-algebra and \(B\) be a large subalgebra of \(A\). In this paper, we show that \(B\) has local weak comparison if \(A\) has local weak comparison, and \(A\) has local weak comparison if \(M_2(B)\) has local weak comparison. As a consequence, we are able to prove that \(A\) has weak comparison if and only if \(B\) has weak comparison.Characterizations of \({*}\) and \({*}\)-left derivable mappings on some algebras.https://www.zbmath.org/1456.460562021-04-16T16:22:00+00:00"An, Guangyu"https://www.zbmath.org/authors/?q=ai:an.guangyu"He, Jun"https://www.zbmath.org/authors/?q=ai:he.jun"Li, Jiankui"https://www.zbmath.org/authors/?q=ai:li.jiankuiSummary: A linear mapping \(\delta\) from a \({*}\)-algebra \(\mathcal{A}\) into a \({*}$-$\mathcal{A} \)-bimodule \(\mathcal{M}\) is a \({*} \)-derivable mapping at \(G\in \mathcal{A}\) if \(A\delta (B)^*+\delta (A)B=\delta (G)\) for each \(A, B\) in \(\mathcal{A}\) with \(AB^*=G\). We prove that every (continuous) \({*} \)-derivable mapping at \(G\) from a (unital \(C^*\)-algebra) factor von Neumann algebra into its Banach \({*}\)-bimodule is a \({*}\)-derivation if and only if \(G\) is a left separating point. A linear mapping \(\delta\) from a \({*}\)-algebra \(\mathcal{A}\) into a \({*} \)-left \(\mathcal{A} \)-module \(\mathcal{M}\) is a \({*} \)-left derivable mapping at \(G\in \mathcal{A}\) if \(A\delta (B)^*+B\delta (A)=\delta (G)\) for each \(A, B\) in \(\mathcal{A}\) with \(AB^*=G\). We prove that every continuous \({*} \)-left derivable mapping at a left separating point from a unital \(C^*\)-algebra or von Neumann algebra into its Banach \({*}\)-left \(\mathcal{A} \)-module is identical with zero under certain conditions.Conditional expectation on non-commutative \(H^{(r,s)}_p(\mathcal{A};\ell_{\infty})\) and \(H_p(\mathcal{A};\ell_1)\) spaces: semifinite case.https://www.zbmath.org/1456.460502021-04-16T16:22:00+00:00"Dauitbek, D."https://www.zbmath.org/authors/?q=ai:dauitbek.dostilek"Tulenov, K."https://www.zbmath.org/authors/?q=ai:tulenov.kanat-serikovichSummary: In this paper, we investigate the conditional expectation on the non-commutative \(H^{(r,s)}_p(\mathcal{A};\ell_{\infty})\) and \(H_p(\mathcal{A};\ell_1)\) spaces associated with semifinite subdiagonal algebra, and prove the contractibility of the underlying conditional expectation on these spaces.Spectral densities of singular values of products of Gaussian and truncated unitary random matrices.https://www.zbmath.org/1456.600262021-04-16T16:22:00+00:00"Neuschel, Thorsten"https://www.zbmath.org/authors/?q=ai:neuschel.thorstenEuler characteristic on noncommutative polyballs.https://www.zbmath.org/1456.460622021-04-16T16:22:00+00:00"Popescu, Gelu"https://www.zbmath.org/authors/?q=ai:popescu.geluSummary: In this paper we introduce and study the Euler characteristic (denoted by \(\chi\)) associated with algebraic modules generated by arbitrary elements of certain noncommutative polyballs. We provide several asymptotic formulas for \(\chi\) and prove some of its basic properties. We show that the Euler characteristic is a complete unitary invariant for the finite rank Beurling type invariant subspaces of the tensor product of full Fock spaces \(F^2(H_{n_1})\otimes\dots\otimes F^2(H_{n_k})\), and prove that its range coincides with the interval \([0,\infty)\). We obtain an analogue of Arveson's version of the Gauss-Bonnet-Chern Theorem from Riemannian geometry, which connects the curvature to the Euler characteristic. In particular, we prove that if \(\mathcal{M}\) is an invariant subspace of \(F^2(H_{n_1})\otimes\dots\otimes F^2(H_{n_k})\), \(n_i\geq2\), which is graded (generated by multi-homogeneous polynomials), then the curvature and the Euler characteristic of the orthocomplement of \(\mathcal{M}\) coincide.Quantum isometries and loose embeddings.https://www.zbmath.org/1456.300982021-04-16T16:22:00+00:00"Chirvasitu, Alexandru"https://www.zbmath.org/authors/?q=ai:chirvasitu.alexandruSummary: We show that countable metric spaces always have quantum isometry groups, thus extending the class of metric spaces known to possess such universal quantum-group actions.
Motivated by this existence problem we define and study the notion of loose embeddability of a metric space \((X,d_X)\) into another, \((Y,d_Y)\): the existence of an injective continuous map that preserves both equalities and inequalities of distances. We show that 0-dimensional compact metric spaces are ``generically'' loosely embeddable into the real line, even though not even all countable metric spaces are.On Popa's factorial commutant embedding problem.https://www.zbmath.org/1456.030582021-04-16T16:22:00+00:00"Goldbring, Isaac"https://www.zbmath.org/authors/?q=ai:goldbring.isaacAuthor's abstract: An open question of Sorin Popa asks whether or not every \(\mathcal{R}^{\mathcal{U}}\)-embeddable factor admits an embedding into \(\mathcal{R}^{\mathcal{U}}\) with factorial relative commutant. We show that there is a locally universal McDuff \(\mathrm{II}_1\) factor \(M\) such that every property (T) factor admits an embedding into \(M^{\mathcal{U}}\) with factorial relative commutant. We also discuss how our strategy could be used to settle Popa's question for property (T) factors if a certain open question in the model theory of operator algebras has a positive solution.
Reviewer: Daniele Mundici (Firenze)Discrete noncommutative Gel'fand Naĭmark duality.https://www.zbmath.org/1456.460612021-04-16T16:22:00+00:00"Bertozzini, Paolo"https://www.zbmath.org/authors/?q=ai:bertozzini.paolo"Conti, Roberto"https://www.zbmath.org/authors/?q=ai:conti.roberto.1"Pitiwan, Natee"https://www.zbmath.org/authors/?q=ai:pitiwan.nateeSummary: We present, in a simplified setting, a non-commutative version of the well-known Gel'fand-Naĭmark duality (between the categories of compact Hausdorff topological spaces and commutative unital \(C^*\)-algebras), where ``geometric spectra'' consist of suitable finite bundles of one-dimensional \(C^*\)-categories equipped with a transition amplitude structure satisfying saturation conditions. Although this discrete duality actually describes the trivial case of finite-dimensional \(C^*\)-algebras, the structures are here developed at a level of generality adequate for the formulation of a general topological/uniform Gel'fand-Naĭmark duality, fully addressed in a companion work.Markovian dynamics under weak periodic coupling.https://www.zbmath.org/1456.812872021-04-16T16:22:00+00:00"Szczygielski, Krzysztof"https://www.zbmath.org/authors/?q=ai:szczygielski.krzysztofSummary: We examine a completely positive and trace preserving evolution of a finite dimensional open quantum system coupled to a large environment via the periodically modulated interaction Hamiltonian. We derive a corresponding Markovian master equation under the usual assumption of weak coupling using the projection operator techniques in two opposite regimes of very small and very large modulation frequency. Special attention is granted to the case of uniformly (globally) modulated interaction, where some general results concerning the Floquet normal form of a solution and its asymptotic stability are also addressed.
{\copyright 2021 American Institute of Physics}Orthogonally additive polynomials on non-commutative \(L^p\)-spaces.https://www.zbmath.org/1456.460512021-04-16T16:22:00+00:00"Alaminos, Jerónimo"https://www.zbmath.org/authors/?q=ai:alaminos.jeronimo"Godoy, María L. C."https://www.zbmath.org/authors/?q=ai:godoy.maria-l-c"Villena, Armando R."https://www.zbmath.org/authors/?q=ai:villena.armando-rThe authors show that, if \(\mathcal{A}\) is a \(C^*\)-algebra (resp., a \(C^*\)-algebra of real rank zero with an increasing approximate unit of projections), \(X\) is a locally convex space (resp., topological vector space), and \(P:\mathcal{A}\to X\) is a continuous \(m\)-homogeneous polynomial, then the following are equivalent: (i) there is a continuous linear mapping \(\Phi:\mathcal{A}\to X\) such that \(P(x)=\Phi(x^m)\) for all \(x\) in \(\mathcal{A}\); (ii) \(P\) is orthogonally additive on \(\mathcal{A}_{sa}\), the self-adjoint part of \(\mathcal{A}\); (iii) \(P\) is orthogonally additive on \(\mathcal{A}_+\), the positive part of \(\mathcal A\).
Using these results, they show that, if \(H\) is a Hilbert space of dimension greater than or equal to \(2\), then there are no non-zero orthogonally additive \(m\)-homogeneous polynomials from \(\mathcal{B}(H)\) into any topological vector space \(X\).
When \(\mathcal{M}\) is a von Neumann algebra with a normal semifinite faithful trace \(\tau\) and \(P\) is a continuous \(m\)-homogeneous polynomial from \(L^p(\mathcal{M},\tau)\) into a topological vector space \(X\), \(0< p<\infty\), the authors show that the following are equivalent: (i) there is a continuous linear mapping \(\Phi: L^{p/m}(\mathcal{M},\tau)\to X\) such that \(P(x)=\Phi(x^m)\) for all \(x\) in \(L^p(\mathcal{M},\tau)\); (ii) \(P\) is orthogonally additive on \(L^p(\mathcal{M},\tau)_{sa}\), the self-adjoint part of \(L^p(\mathcal{M},\tau)\); (iii) \(P\) is orthogonally additive on \(S(\mathcal{M},\tau)_+\).
This allows the authors to present a number of results concerning orthogonally additive polynomials on the Schatten \(p\)-class, \(S^p(H)\), and the space of compact operators on a Hilbert space \(H\), \(\mathcal{K}(H)\). From this it follows that, if \(H\) is a Hilbert space of dimension at least \(2\), then for \(0< p<\infty\), there are no non-zero orthogonally additive polynomials on \(S^p(H)\).
Reviewer: Christopher Boyd (Dublin)Aharonov-Bohm superselection sectors.https://www.zbmath.org/1456.812972021-04-16T16:22:00+00:00"Dappiaggi, Claudio"https://www.zbmath.org/authors/?q=ai:dappiaggi.claudio"Ruzzi, Giuseppe"https://www.zbmath.org/authors/?q=ai:ruzzi.giuseppe"Vasselli, Ezio"https://www.zbmath.org/authors/?q=ai:vasselli.ezioSummary: We show that the Aharonov-Bohm effect finds a natural description in the setting of QFT on curved spacetimes in terms of superselection sectors of local observables. The extension of the analysis of superselection sectors from Minkowski spacetime to an arbitrary globally hyperbolic spacetime unveils the presence of a new quantum number labelling charged superselection sectors. In the present paper, we show that this ``topological'' quantum number amounts to the presence of a background flat potential which rules the behaviour of charges when transported along paths as in the Aharonov-Bohm effect. To confirm these abstract results, we quantize the Dirac field in the presence of a background flat potential and show that the Aharonov-Bohm phase gives an irreducible representation of the fundamental group of the spacetime labelling the charged sectors of the Dirac field. We also show that non-abelian generalizations of this effect are possible only on spacetimes with a non-abelian fundamental group.