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

2015/05/25 by Huichi Huang, Huang, Huichi, Jianchao Wu +1
Mathematics · #37B99 #46L30 #46L55 #Dynamical Systems (math.DS) #FOS: Mathematics #Operator Algebras (math.OA) #math.DS #math.OA #msc:37B99 #msc:46L30 #msc:46L55

paper · pdf · doi:10.48550/arxiv.1505.06633

Introduction rewritten, 20 pages. Comments welcome

arxiv created 2016/02/26 · arxiv updated 2016/03/01

Abstract

Motivated by reformulating Furstenberg's × p,× q conjecture via representations of a crossed product C^*-algebra, we show that in a discrete C^*-dynamical system (A,Γ), the space of (ergodic) Γ-invariant states on A is homeomorphic to a subspace of (pure) state space of A\rtimesΓ. Various applications of this in topological dynamical systems and representation theory are obtained. In particular, we prove that the classification of ergodic Γ-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 Γ.

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