On Reduced Amalgamated Free Products of C*-algebras and the MF-Property

TL;DR

This paper proves the MF property for reduced group C*-algebras of amalgamated free products of Abelian groups and characterizes when their BDF semigroup forms a group, with applications to tensor product factorizations.

Contribution

It establishes the MF property for these algebras and provides a characterization of when their BDF semigroup is a group, including new examples.

Findings

MF property holds for reduced group C*-algebras of amalgamated free products of Abelian groups.

Characterization of when the BDF semigroup of these algebras is a group.

Tensor product factorization of the associated group von Neumann algebra.

Abstract

We establish the MF property of the reduced group $ C^* $-algebra of an amalgamated free product of countable Abelian discrete groups. This result is then used to give a characterization of the amalgamated free products of Abelian groups for which the BDF semigroup of the reduced group $ C^* $-algebra is a group. Along the way we get a tensor product factorization of the corresponding group von Neumann algebra. We end the exposition by applying the ideas from the first part to give a few more examples of groups with a reduced group $ C^* $-algebra which is MF.