# Positive Herz-Schur multipliers and approximation properties of crossed products

## TL;DR

This paper characterizes positive Herz-Schur multipliers on crossed product $C^*$-algebras and links them to approximation properties like nuclearity and the Haagerup property, extending previous work on their structure and applications.

## Contribution

It provides a Stinespring-type characterization of positive Schur $A$-multipliers and connects them to approximation properties of crossed products, advancing understanding of their structure.

## Key findings

- Characterization of positive Herz-Schur multipliers via Stinespring-type theorem.
- Relation of these multipliers to approximation properties such as nuclearity.
- Implementation of approximation properties in crossed products through these multipliers.

## Abstract

For a $C^*$-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 $C^*$-algebraic crossed products of the form $A\rtimes_{\alpha,r} G$, with $G$ a discrete group, whose various versions were considered earlier by Anantharaman-Delaroche, B\'edos and Conti, and Dong and Ruan. The latter maps are shown to implement approximation properties, such as nuclearity or the Haagerup property, for $A\rtimes_{\alpha,r} G$.