Quasi-invariant states for the action of compact groups

TL;DR

This paper investigates G-quasi-invariant states in C*-algebras under compact group actions, showing that under certain conditions, these states' GNS representations are equivalent to G-invariant states, thus deepening understanding of symmetry in operator algebras.

Contribution

It establishes a link between G-quasi-invariant states with central support and G-invariant states when the group action commutes with the modular group.

Findings

G-quasi-invariant states with central support have GNS representations equivalent to G-invariant states under specific conditions.

The paper provides conditions under which the automorphic action of a compact group leads to equivalence of representations.

It advances the understanding of symmetry properties in the context of C*-algebraic states and group actions.

Abstract

We analyze a natural C*-algebraic definition of G-quasi-invariant states for the automorphic action of a compact group G. We prove that, given a G-quasi-invariant state with central support, when the action of the group G commutes with the modular group, its GNS representation is equivalent to that of a G-invariant state.