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

Abstract

Given a C∗-dynamical system consisting of a unital C∗-algebra A and a discrete group Γ acting on A as automorphisms, we show that the space of (ergodic) Γ-invariant states on A is homeomorphic to a subspace of (pure) state space of A o Γ. It follows that classification of ergodic Γ-invariant regular Borel probability measures on a compact Hausdorff space X is equivalent to classification of irreducible representations of C(X) o Γ whose restriction to Γ contains the trivial representation of Γ. As an application, a reformulation of Furstenberg’s conjecture via representation theory of the semidirect product group Z[ 1 pq ] o Z 2 is obtained.