QUASI-INVARIANT STATES FOR COMPACT GROUP ACTIONS: REDUCTION TO NORMAL SUBGROUPS AND QUOTIENTS

Abstract

We establish a structural decomposition theorem for quasi-invariant states under compact group actions. For a compact group G , a closed normal subgroup H and an action

on a separable

C ∗ $C^*$ upper C Superscript asterisk

-algebra

A $\mathfrak {A}$ German upper A

, we prove that a state

ω $\omega $ omega

with central support, for which the lifted action commutes with the modular group, is G -quasi-invariant if and only if it is H -quasi-invariant and its restriction to the fixed-point algebra

A H $\mathfrak {A}^H$ German upper A Superscript upper H

is

G / H $G/H$ upper G divided by upper H

-quasi-invariant. This completely characterises quasi-invariance through normal subgroup data. The proof uses modular theory, conditional expectations and the theory of correspondences. Applications to classical quasi-invariant measures and to the CAR algebra are discussed.