The MF property for amalgamated free products

TL;DR

This paper investigates the MF property in C*-algebras and groups, establishing conditions under which amalgamated free products retain the MF property and identifying classes of groups with MF full group C*-algebras.

Contribution

It proves that amalgamated free products of MF C*-algebras are MF and characterizes when general amalgamated free products are MF, also identifying several groups with MF full group C*-algebras.

Findings

Amalgamated free products of MF C*-algebras are MF.

Necessary and sufficient conditions for an amalgamated free product to be MF.

Certain groups, including amalgamated free products of amenable groups, have MF full group C*-algebras.

Abstract

A C*-algebra (or a group) is called MF (matricial field) if it admits finite dimensional approximate unitary representations which are approximately injective, where approximately is meant with respect to the operator norm. It is proved that for any MF C*-algebra $A$ and its C*-subalgebra $C$, $A\ast_C A$ is MF. For general amalgamated free products, $A\ast_C B$, a necessary and sufficient condition for being MF is given. It is shown that the following groups -- amalgamated free products of amenable groups, semidirect products of amenable groups by free groups, and $\mathbb Z^2\rtimes SL_2(\mathbb Z)$ -- all have MF full group C*-algebra. It is shown that the class of MF C*-algebras is closed under maximal tensor products with $C^*(\mathbb F_n)$.