Crossed products of nuclear \(\mathrm C^{\ast}\)-algebras and their traces
type: Item
instance of: scholarly article
title: Crossed products of nuclear \(\mathrm C^{\ast}\)-algebras and their traces
zbMATH Open document ID: 1473.46069
author: Timothy Rainone
author: Christopher P. Schafhauser
published in: Advances in Mathematics
publication date: 2019-03-28T00:00:00Z
review text: This interesting paper studies the Blackadar-Kirchberg Question (BKQ), which is a C*-analogue of the Connes embedding conjecture for von Neumann algebras. More precisely, it is investigated when certain C*-algebras are matricial field (MF), a property that asks when the C*-algebra embeds into a norm ultraproduct of matrix algebras. The BKQ asks whether every stably finite C*-algebra is MF. The main results of the paper prove that certain crossed products or group C*-algebras are MF. They also deal with tracial versions of the MF and prove results in this direction for the same classes of C*-algebras. A characterization is given of when a (reduced) crossed product by a free group is MF. This characterization can be phrased both in terms of stable finiteness or in terms of $K$-theory. In particular, it is proved that a crossed product of the form \(A\rtimes_r \mathbb{F}_r\) is MF whenever \(A\) is separable, simple, unital, nuclear and monotracial satisfying the UCT and \(\mathbb{F}_r\) is a free group of rank~\(r\). The (tracial) MF property is also investigated for (reduced) C*-algebras of certain semidirected products of groups of the form \(G\rtimes \mathbb{F}_r\) for an amenable group \(G\) or \(H\rtimes G\) for a locally finite group \(G\) and a group f\(H\) or which \(C^*_r(H)\) has the MF property.
reviewed by: Alcides Buss
Mathematics Subject Classification ID: 46L35
Mathematics Subject Classification ID: 46L55
zbMATH DE Number: 7044289
zbMATH Keywords: MF algebras
zbMATH Keywords: quasidiagonality
zbMATH Keywords: crossed products
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: dynamical systems
MaRDI profile type: Publication
full work available at URL: j.aim.2019.01.045
OpenAlex ID: W2916699178
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DOI: 10.1016/J.AIM.2019.01.045
description: scientific article; zbMATH DE number 7044289