Covariant Representations of C*-algebras and their Compact Automorphism Groups

TL;DR

This paper explores the structure of covariant representations of C*-algebras under compact group actions, establishing conditions for transitivity and ergodicity, and demonstrating that irreducible representations are induced from stability groups.

Contribution

It proves that group actions are transitive if and only if ergodic on standard Borel G-measure spaces, and shows all irreducible covariant representations are induced from stability groups.

Findings

Group action transitivity is equivalent to ergodicity.

Irreducible covariant representations are induced from stability groups.

The system satisfies strong-EHI property.

Abstract

Let G be a compact group. Let (X,G) be a standard Borel G-measure space. We show that the group action on (X, G) is transitive if and only if it is ergodic. Using this result, we show that every irreducible covariant representation of a C*-dynamical system (A, G) is induced from a stability group. In addition, we show that (A, G) satisfies strong-EHI.