Exactness and SOAP of crossed products via Herz-Schur multipliers. (English) Zbl 1459.42014
Summary: Given a \(C^\ast \)-dynamical system \((A, G, \alpha)\), with \(G\) a discrete group, Schur \(A\)-multipliers and Herz-Schur \((A, G, \alpha)\)-multipliers are used to implement approximation properties, namely exactness and the strong operator approximation property (SOAP), of \(A \rtimes_{\alpha , r} G\). The resulting characterisations of exactness and SOAP of \(A \rtimes_{\alpha , r} G\) generalise the corresponding statements for the reduced group \(C^\ast \)-algebra.
MSC:
| 42B15 | Multipliers for harmonic analysis in several variables |
| 46L55 | Noncommutative dynamical systems |
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