Semigroups of Herz-Schur Multipliers

TL;DR

This paper explores the connection between approximation properties of groups and semigroups of Herz-Schur multipliers, introducing the weak Haagerup property and analyzing their generators, especially in free groups.

Contribution

It introduces the weak Haagerup property and characterizes generators of Herz-Schur multiplier semigroups, linking them to group properties like amenability.

Findings

Weak Haagerup property is equivalent to semigroup existence.

Generators split into positive definite and conditionally negative definite kernels.

Generators on free groups are linearly bounded by word length.

Abstract

In order to investigate the relationship between weak amenability and the Haagerup property for groups, we introduce the weak Haagerup property, and we prove that having this approximation property is equivalent to the existence of a semigroup of Herz-Schur multipliers generated by a proper function. It is then shown that a (not necessarily proper) generator of a semigroup of Herz-Schur multipliers splits into a positive definite kernel and a conditionally negative definite kernel. We also show that the generator has a particularly pleasant form if and only if the group is amenable. In the second half of the paper we study semigroups of radial Herz-Schur multipliers on free groups. We prove that a generator of such a semigroup is linearly bounded by the word length function.