Abstract
Given a C⁎-dynamical system (A,G,α), with G a discrete group, Schur A-multipliers and Herz–Schur (A,G,α)-multipliers are used to implement approximation properties, namely exactness and the strong operator approximation property (SOAP), of A⋊α,rG. The resulting characterisations of exactness and SOAP of A⋊α,rG generalise the corresponding statements for the reduced group C⁎-algebra.
Highlights
In [16] the notion of classical Schur multipliers, which has been intensively studied in the literature, was generalised to the operator-valued setting: for a C∗-algebra A ⊂ B(H) on a Hilbert space H, and a set X, Schur A-multipliers were defined as functions φ : X × X → CB(A, B(H)) such that the associated map Sφ : K(l2(X)) ⊗ A → K(l2(X)) ⊗ B(H) is completely bounded
Herz–Schur multipliers for the reduced crossed product were defined in [4] for discrete groups and in [16] for general locally compact groups; in the latter they were related to Schur A-multipliers in a similar manner as for the group case
In the Corollary below we provide further evidence that Herz–Schur multipliers can be used as a technical basis for approximation results by using Herz–Schur multipliers to prove the corresponding exactness result [5, Theorem 4.3.4 (3)] and its generalisation
Summary
In [16] the notion of classical Schur multipliers, which has been intensively studied in the literature (see e.g. [11, 13, 18, 19]), was generalised to the operator-valued setting: for a C∗-algebra A ⊂ B(H) on a Hilbert space H, and a set X, Schur A-multipliers were defined as functions φ : X × X → CB(A, B(H)) such that the associated map Sφ : K(l2(X)) ⊗ A → K(l2(X)) ⊗ B(H) is completely bounded. Schur multipliers are related to the notion of Herz–Schur multipliers associated to groups The latter are functions ψ : G → C on a locally compact group G that give rise to completely bounded maps on the reduced C∗algebra. We finish with some observations about duality for the space of Herz–Schur multipliers and a certain commutative subspace thereof
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