Positive Herz-Schur multipliers and approximation properties of crossed products

Abstract

For a $C^{\ast }$-algebra $A$ and a set $X$ we give a Stinespring-type characterisation of the completely positive Schur $A$-multipliers on $K({\ell }^2(X))\otimes A$. We then relate them to completely positive Herz-Schur multipliers on $C^{\ast }$-algebraic crossed products of the form $A{⋊}_{\alpha ,r}G$, with $G$ a discrete group, whose various versions were considered earlier by Anantharaman-Delaroche, Bédos and Conti, and Dong and Ruan. The latter maps are shown to implement approximation properties, such as nuclearity or the Haagerup property, for $A{⋊}_{\alpha ,r}G$.

21 pages, v2 corrects a few minor typos. The paper will appear in the Mathematical Proceedings of the Cambridge Philosophical Society

References in corpus (1)

• Herz-Schur multipliers of dynamical systems

Cited by in corpus (4)

• Amenability and weak containment for actions of locally compact groups on $C^{\ast }$-algebras
• Positive definiteness and Fell bundles over discrete groups
• Amenability, Nuclearity and Tensor Products of $C^{\ast }$-Algebraic Fell Bundles under the Unified Viewpoint of the Fell-Doran Induced Representation Theory
• Central and convolution Herz-Schur multipliers