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QUASI-INVARIANT STATES FOR COMPACT GROUP ACTIONS: REDUCTION TO NORMAL SUBGROUPS AND QUOTIENTS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 24 September 2026, pp. 1-10
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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 $\alpha :G\to \mathrm {Aut}(\mathfrak {A})$
on a separable $C^*$
-algebra $\mathfrak {A}$
, we prove that a state $\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 $\mathfrak {A}^H$
is $G/H$
-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.
