Ergodic invariant states and irreducible representations of crossed product $C^*$-algebras

TL;DR

This paper explores the relationship between ergodic invariant states and irreducible representations in crossed product $C^*$-algebras, with applications to dynamical systems and representation theory, inspired by Furstenberg's conjecture.

Contribution

It establishes a homeomorphism between ergodic invariant states and a subset of pure states of crossed product $C^*$-algebras, linking measure classification to irreducible representations.

Findings

Homeomorphism between ergodic invariant states and pure states of crossed product $C^*$-algebras

Application to classification of invariant measures on compact spaces

Connections between dynamical systems and irreducible representations

Abstract

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. 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 $C(X)\rtimes \Gamma$.