Simple reduced $L^p$ operator crossed products with unique trace

Abstract

In this article we study simplicity and traces of reduced $L^p$ operator crossed products $F_{\mathrm{r}}^p(G,A,\alpha )$. Given $p\in (1,\infty )$, let $G$ be a Powers group, and let $\alpha :G\to Aut(A)$ be an isometric action of $G$ on a unital $L^p$ operator algebra $A$ such that $A$ is $G$-simple. We prove that the reduced $L^p$ operator crossed product of $A$ by $G$, $F_{\mathrm{r}}^p(G,A,\alpha )$, is simple. Moreover, we show that traces on $F_{\mathrm{r}}^p(G,A,\alpha )$ are in correspondence with $G$-invariant traces on A. Our results generalize the results obtained by de la Harpe for reduced $C^{\ast }$crossed products in 1985. By letting $G$ be a countable nonabelian free group as a special case, we recover an analogue of a result of Powers from 1975. For the case $p=1$, it turns out that (reduced) $L^p$ operator group algebras are not simple.

12 pages

References in corpus (1)

• $C^{\ast }$-simple groups without free subgroups

Cited by in corpus (2)

• Functoriality of group algebras acting on $L^p$-spaces
• Representations of étale groupoids on $L^p$-spaces