A remark on the freeness condition of Suzuki’s correspondence theorem for intermediate \(C^*\)-algebras. (English) Zbl 1483.46058
Summary: Let \(\Gamma\) be a discrete group satisfying the approximation property (AP). Let \(X, Y\) be \(\Gamma \)-spaces and \(\pi : Y \to X\) be a proper factor map which is injective on the non-free part. We prove the one-to-one correspondence between intermediate \(\mathrm{C}^*\)-algebras of \(C_0(X) \rtimes_r \Gamma \subset C_0(Y) \rtimes_r \Gamma\) and intermediate \(\Gamma\)-\(\mathrm{C}^\ast \)-algebras of \(C_0(X) \subset C_0(Y)\). This is a generalization of Suzuki’s theorem that proves the statement for free actions.
MSC:
| 46L05 | General theory of \(C^*\)-algebras |
References:
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