Abstract

Herz—Schur multipliers of a discrete group have proved useful in the study of $C^{$-algebras, as they can be used to link properties of the group to approximation properties of the reduced group $C}$-algebra. The development of these ideas has shed light on some $C<sup>$-algebra properties, and motivated the introduction and study of others.

I will introduce Herz—Schur multipliers, and discuss some of their applications, before describing a generalisation of these functions to multipliers of a $C</sup>$-dynamical system. In the final part of the talk I will show how the generalised Herz—Schur multipliers can be used to study approximation properties of the reduced crossed product formed by a group acting on a $C^*$-algebra, paralleling the applications of Herz—Schur multipliers; these new tools allow us to study approximation properties of the reduced crossed product without requiring the group to be amenable.