Journal of Operator Theory

Volume 74, Issue 1, Summer 2015 pp. 133-147.

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

Authors: (1) Shirin Hejazian, (2) Sanaz Pooya
Author institution: (1) Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad 91775, Iran
(2) Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad 91775, Iran

Summary: Given $p \in (1, \infty)$, let $G$ be a countable Powers group, and let $(G, A, \alpha)$ be a separable nondegenerately representable isometric $G$-$L^p$-operator algebra. We show that if $A$ is unital and $G$-simple then the reduced $L^p$-operator crossed product of $A$ by $G$, $F^p_{\mathrm{r}}(G, A, \alpha)$, is simple. Furthermore, traces on $F^p_{\mathrm{r}}(G, A, \alpha)$ are in natural bijection with $G$-invariant traces on $A$ via the standard conditional expectation. In particular, if $A$ has a unique normalized trace then so does $F^p_{\mathrm{r}}(G, A, \alpha)$. These results generalize special cases of some results due to de la Harpe and Skanadalis in the case of $C^*$-algebras.

DOI: http://dx.doi.org/10.7900/jot.2014may13.2036
Keywords: $L^p$-operator algebra, $G$-$L^p$-operator algebra, covariant representation, regular covariant representation, crossed product, Powers group, $G$-invariant ideal, simple algebra, trace

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