Crossed products of UHF algebras by some amenable groups
type: Item
instance of: scholarly article
title: Crossed products of UHF algebras by some amenable groups
zbMATH Open document ID: 0962.46047
published in: Hokkaido Mathematical Journal
publication date: 2001-05-30T00:00:00Z
review text: Let \( C^* \)-algebra \( A \) be UHF and \( \alpha:G\to \text{ Aut}(A) \) a representation of an amenable group \( G \). The principal result is that if \( G=\mathbb Z^n \), there exists a \(*\)-monomorphism \( p:A\rtimes_\alpha\mathbb Z^n\to B\) where \( B \) is AF (i.e., the crossed product is AF embeddable) and \( \rho_*: K_0(A\rtimes_\alpha\mathbb Z^n)\to K_0(B) \) is also injective. Together with an imprimitivity theorem of \textit{P. Green} [see Cor. 2.8. J. Funct. Anal. 36, 88-104 (1980; Zbl 0422.46048)] this entails a generalization. If \( G\in\Gamma \) (resp. \( G\in\Gamma_{fg} \)) then \( A\rtimes_\alpha G \) is quasidiagonal (resp. AF-embeddable). Here \( \Gamma \) (resp. \( \Gamma_{fg} \)) is the class of separable l.c. groups \( G \) which have a finite series of subgroups \( H_1\subset H_2\subset\dots\subset H_n=G \) such that \( H_1 \) is discrete and abelian (resp., moreover, finitely generated) and, for \( i=1,\dots,n-1 \), \( H_i \) is a closed normal subgroup of \( H_{i+1} \) with \( H_{i+1}/H_i \) compact. The proof relies on the work by \textit{A. Kishimoto} [``Automorphisms of \(C^*\)-algebras with the Rohlin property'', J. Oper. Theory]. The main tools are \( K \)-theory and the Rohlin property of automorphisms.
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 46L80
zbMATH DE Number: 1448536
zbMATH Keywords: UHF \(C^*\)-algebra
zbMATH Keywords: quasidiagonality
zbMATH Keywords: crossed products
zbMATH Keywords: amenable groups
zbMATH Keywords: Rohlin property of automorphisms
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: AF embedding
author: Nathanial P. Brown
reviewed by: Andrey V. Bulinski
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1350912964
OpenAlex ID: W2069273082
DOI: 10.14492/HOKMJ/1350912964
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description: scientific article; zbMATH DE number 1448536