Ergodic invariant states and irreducible representations of crossed product $C^{\ast }$-algebras

Abstract

Motivated by reformulating Furstenberg's $\times p,\times q$ conjecture via representations of a crossed product $C^{\ast }$-algebra, we show that in a discrete $C^{\ast }$-dynamical system $(A,\Gamma )$, the space of (ergodic) $\Gamma$-invariant states on $A$ is homeomorphic to a subspace of (pure) state space of $A⋊\Gamma$. Various applications of this in topological dynamical systems and representation theory are obtained. In particular, we prove that the classification of ergodic $\Gamma$-invariant regular Borel probability measures on a compact Hausdorff space $X$ is equivalent to the classification a special type of irreducible representations of $C(X)⋊\Gamma$.

arXiv commentIntroduction rewritten, 20 pages. Comments welcome