A Schur multiplier characterization of coarse embeddability

TL;DR

This paper characterizes when locally compact groups can be coarsely embedded into Hilbert spaces using Schur multipliers, linking this property to the weak Haagerup constant and implications for the Baum-Connes conjecture.

Contribution

It introduces a Schur multiplier criterion for coarse embeddability of groups into Hilbert spaces, connecting harmonic analysis with geometric group theory.

Findings

Groups with weak Haagerup constant 1 embed coarsely into Hilbert spaces

Coarse embeddability implies split-injectivity of Baum-Connes assembly map with coefficients

Provides a new analytical tool for studying geometric properties of groups

Abstract

We give a contractive Schur multiplier characterization of locally compact groups coarsely embeddable into Hilbert spaces. Consequently, all locally compact groups whose weak Haagerup constant is 1 embed coarsely into Hilbert spaces, and hence the Baum-Connes assembly map with coefficients is split-injective for such groups.