Journal of Operator Theory
Volume 74, Issue 1, Summer 2015 pp. 133-147.
Simple reduced $L^p$-operator crossed products with
unique trace
Authors:
Shirin Hejazian (1) and Sanaz Pooya (2)
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 $G$-$L^p$-opera\-tor
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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