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September 2020 Subalgebras of simple AF-algebras
Christopher Schafhauser
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Ann. of Math. (2) 192(2): 309-352 (September 2020). DOI: 10.4007/annals.2020.192.2.1

Abstract

It is shown that if $A$ is a separable, exact $C^*$-algebra which satisfies the Universal Coefficient Theorem (UCT) and has a faithful, amenable trace, then A admits a trace-preserving embedding into a simple, unital AF-algebra with a unique trace. Modulo the UCT, this provides an abstract characterization of $C^*$-sub­algebras of simple, unital AF-algebras.

As a consequence, for a countable, discrete, amenable group $G$ acting on a second countable, locally compact, Hausdorff space $X$, $C_0(X) \rtimes_r G$ embeds into a simple, unital AF-algebra if, and only if, $X$ admits a faithful, invariant, Borel, probability measure. Also, for any countable, discrete, amenable group $G$, the reduced group $C^*$-algebra $C^*_r(G)$ admits a trace-preserving embedding into the universal UHF-algebra.

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Christopher Schafhauser. "Subalgebras of simple AF-algebras." Ann. of Math. (2) 192 (2) 309 - 352, September 2020. https://doi.org/10.4007/annals.2020.192.2.1

Information

Published: September 2020
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2020.192.2.1

Subjects:
Primary: 46L05

Keywords: AF-embedding , amenable trace

Rights: Copyright © 2020 Department of Mathematics, Princeton University

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Vol.192 • No. 2 • September 2020
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