Journal of Operator Theory
Volume 78, Issue 1, Summer 2017 pp. 159-172.
Ergodic invariant states and irreducible
representations of crossed product $C^*$-algebras
Authors:
Huichi Huang (1) and Jianchao Wu (2)
Author institution: (1) College of Mathematics and Statistics, Chongqing
University, Chongqing, 401331, China
(2) Department of Mathematics, Pennsylvania State University, University
park, Pennsylvania, 16802, U.S.A.
Summary: Motivated by reformulating Furstenberg's $\times
p,\times q$ conjecture via
representations of a crossed product $C^*$-algebra, we show that in a
discrete $C^*$-dynamical system $(A,\Gamma)$, the space of (ergodic)
$\Gamma$-invariant states on $A$ is homeomorphic to a subspace of (pure)
state
space of $A\rtimes\Gamma$. Various applications of this in topological
dynamical systems and representation theory are obtained.
DOI: http://dx.doi.org/10.7900/jot.2016jun11.2141
Keywords: invariant state, crossed product $C^*$-algebra,
irreducible representation
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