Unital Full Amalgamated Free Products of MF Algebras

TL;DR

This paper investigates conditions under which unital full free products of MF algebras with amalgamation over finite dimensional C*-algebras remain MF algebras, extending known results for RFD and AF algebras.

Contribution

It establishes that under natural conditions, free products of MF algebras and AF algebras with amalgamation preserve the MF property.

Findings

Unital full free products of two RFD C*-algebras with amalgamation are RFD.

Under certain conditions, free products of MF algebras are MF.

Free products of AF algebras with compatible faithful tracial states are MF.

Abstract

In this paper, we consider the question whether a unital full free product of MF algebras with amalgamation over a finite dimensional C*-algebra is an MF algebra. First, we show that, under a natural condition, a unital full free product of two separable residually finite dimensional (RFD) C*-algebras with amalgamation over a finite dimensional C*-algebra is again a separable RFD C*-algebra. Applying this result on MF C*-algebras, we show that, under a natual condition, a unital full free product of two MF algebras is again an MF algebra. As an application, we show that a unital full free product of two AF algebras with amalgamation over an AF algebra is an MF algebra if there are faithful tracial states on each of these two AF algebras such that the restrictions on the common subalgebra agree.