Abstract
We extend a theorem of Haagerup and Kraus in the $\mathrm{C}^{\ast}$-algebra context: for a locally compact group with the approximation property (AP), the reduced $\mathrm{C}^{\ast}$-crossed product construction preserves the strong operator approximation property (SOAP). In particular their reduced group $\mathrm{C}^{\ast}$-algebras have the SOAP. Our method also solves another open problem: the AP implies exactness for general locally compact groups.
Funding Statement
This work was supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (Start-up, No. 17H06737) and tenure track funds of Nagoya University.
Citation
Yuhei SUZUKI. "The approximation property and exactness of locally compact groups." J. Math. Soc. Japan 73 (1) 263 - 275, January, 2021. https://doi.org/10.2969/jmsj/83368336
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