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The Cuntz semigroup and the radius of comparison of the crossed product by a finite group

Part of: $K$-theory and operator algebras Selfadjoint operator algebras

Published online by Cambridge University Press:  10 March 2021

M. ALI ASADI-VASFI ,
NASSER GOLESTANI  and
N. CHRISTOPHER PHILLIPS
M. ALI ASADI-VASFI
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR97403-1222, USA School of Mathematics, Statistics, and Computer Science, University of Tehran, Tehran, Iran
NASSER GOLESTANI*
Affiliation:
Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, P. O. Box 14115–134, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395–5746, Tehran, Iran
N. CHRISTOPHER PHILLIPS
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR97403-1222, USA
*
n.golestani@modares.ac.ir
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Abstract

Let G be a finite group, let A be an infinite-dimensional stably finite simple unital C*-algebra, and let $\alpha \colon G \to {\operatorname {Aut}} (A)$ be an action of G on A which has the weak tracial Rokhlin property. Let $A^{\alpha }$ be the fixed point algebra. Then the radius of comparison satisfies ${\operatorname {rc}} (A^{\alpha }) \leq {\operatorname {rc}} (A)$ and ${\operatorname {rc}} ( C^* (G, A, \alpha ) ) \leq ({1}/{{{\operatorname{card}}} (G))} \cdot {\operatorname {rc}} (A)$ . The inclusion of $A^{\alpha }$ in A induces an isomorphism from the purely positive part of the Cuntz semigroup ${\operatorname {Cu}} (A^{\alpha })$ to the fixed points of the purely positive part of ${\operatorname {Cu}} (A)$ , and the purely positive part of ${\operatorname {Cu}} ( C^* (G, A, \alpha ) )$ is isomorphic to this semigroup. We construct an example in which $G \,{=}\, {\mathbb {Z}} / 2 {\mathbb {Z}}$ , A is a simple unital AH algebra, $\alpha $ has the Rokhlin property, ${\operatorname {rc}} (A)> 0$ , ${\operatorname {rc}} (A^{\alpha }) = {\operatorname {rc}} (A)$ , and ${\operatorname {rc}} ({C^* (G, A, \alpha)} ) = ({1}/{2}) {\operatorname {rc}} (A)$ .

Keywords

simple C*-algebraradius of comparisoncrossed productweak tracial Rokhlin property

MSC classification

Primary: 46L55: Noncommutative dynamical systems
Secondary: 19K14: $K_0$ as an ordered group, traces 46L80: $K$-theory and operator algebras (including cyclic theory)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 12 , December 2021 , pp. 3541 - 3592
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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