Recent zbMATH articles in MSC 19Lhttps://www.zbmath.org/atom/cc/19L2021-04-16T16:22:00+00:00WerkzeugClassification 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)Elliptic classes of Schubert varieties.https://www.zbmath.org/1456.140602021-04-16T16:22:00+00:00"Kumar, Shrawan"https://www.zbmath.org/authors/?q=ai:kumar.shrawan"Rimányi, Richárd"https://www.zbmath.org/authors/?q=ai:rimanyi.richard"Weber, Andrzej"https://www.zbmath.org/authors/?q=ai:weber.andrzejSummary: We introduce new notions in elliptic Schubert calculus: the (twisted) Borisov-Libgober classes of Schubert varieties in general homogeneous spaces \(G/P\). While these classes do not depend on any choice, they depend on a set of new variables. For the definition of our classes we calculate multiplicities of some divisors in Schubert varieties, which were only known for full flag varieties before. Our approach leads to a simple recursions for the elliptic classes. Comparing this recursion with R-matrix recursions of the so-called elliptic weight functions of Rimanyi-Tarasov-Varchenko we prove that weight functions represent elliptic classes of Schubert varieties.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)