Kirchberg's factorization property for locally compact groups

TL;DR

This paper explores Kirchberg's factorization property for locally compact groups, establishing new conditions and extending known characterizations from discrete groups to a broader class.

Contribution

It provides a partial solution to the passing of the factorization property to embeddings and extends Kirchberg's characterization to residually amenably embeddable groups.

Findings

Residually amenably embeddable groups have the factorization property

A partial answer to the inheritance of the property under embeddings

Extension of Kirchberg's characterization to a broader class of groups

Abstract

A locally compact group $G$ has the factorization property if the map $$C^*(G)\odot C^*(G)\ni a\otimes b\mapsto \lambda(a)\rho(b)\in\mathcal B(L^2(G))$$ is continuous with respect to the minimal C*-norm. This paper seeks to initiate a rigorous study of this property in the case of locally compact groups which, in contrast to the discrete case, has been relatively untouched. A partial solution to the question of when the factorization property passes to continuous embeddings is given -- a question which traces back to Kirchberg's seminal work on the topic and is known to be false in general. It is also shown that every "residually amenably embeddable" group must necessarily have the factorization property and that an analogue of Kirchberg's characterization of the factorization property for discrete groups with property (T) holds for a more general class of groups.