A remark on the freeness condition of Suzuki's correspondence theorem for intermediate C$^*$-algebras

Abstract

Let $\Gamma$ be a discrete group satisfying the approximation property (AP). Let $X$, $Y$ be $\Gamma$-spaces and $\pi \colon Y \to X$ be a proper factor map which is injective on the non-free part. We prove the one-to-one correspondence between intermediate C$^*$-algebras of $C_0(X) \rtimes_r \Gamma \subset C_0(Y) \rtimes_r \Gamma$ and intermediate $\Gamma$-${\rm C}^\ast$-algebras of $C_0(X) \subset C_0(Y)$. This is a generalization of Suzuki's theorem that proves the statement for free actions.