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

Abstract

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 α : G → Aut ( A ) $\alpha :G\to \mathrm {Aut}(\mathfrak {A})$ alpha colon upper G right arrow upper A u t left parenthesis German upper A right parenthesis 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.

The authors' abstract, as published at the source. Bulletin of the Australian Mathematical Society, 2026 · DOI ↗