# 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