# A remark on the freeness condition of Suzuki's correspondence theorem for intermediate ${\rm C}^\ast$-algebras

## TL;DR

This paper generalizes Suzuki's correspondence theorem for intermediate C*-algebras to non-free actions of groups with the approximation property, establishing a one-to-one correspondence under certain conditions.

## Contribution

It extends Suzuki's theorem to non-free actions by proving a correspondence between intermediate C*-algebras for group actions with the AP.

## Key findings

- Established a bijective correspondence for intermediate C*-algebras in non-free actions.
- Generalized Suzuki's theorem beyond free actions.
- Applicable to groups satisfying the approximation property.

## 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 ${\rm C}^\ast$-algebras of $C_0(X) \rtimes_r \Gamma \subset C_0(Y) \rtimes \Gamma$ 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.