Title: Positive Herz-Schur multipliers and approximation properties of crossed products,
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, Bédos and Conti, and Dong and
Ruan. The latter maps are shown to implement approximation properties, such as
nuclearity or the Haagerup property, for $A\rtimes_{\alpha,r} G$.