Central amalgamation of groups and the RFD property

TL;DR

This paper explores when full group C*-algebras are residually finite-dimensional, extending known results for central amalgamated free products of virtually abelian groups to broader classes, and providing counterexamples to closure properties.

Contribution

It generalizes the class of groups for which central amalgamated free products are RFD and shows that the class of RFD C*-algebras is not closed under such constructions.

Findings

Central amalgamated free products of polycyclic-by-finite groups over finitely generated central subgroups are RFD.

The class of RFD C*-algebras is not closed under central amalgamated free products.

Counterexamples of RFD groups whose central amalgamated free product is not RFD.

Abstract

It is an old and challenging topic to investigate for which discrete groups G the full group C*-algebra C*(G) is residually finite-dimensional (RFD). In particular not much is known about how the RFD property behaves under fundamental constructions, such as amalgamated free products and HNN-extensions. In [CS19] it was proved that central amalgamated free products of virtually abelian groups are RFD. In this paper we prove that this holds much beyond this case. Our method is based on showing a certain approximation property for characters induced from central subgroups. In particular it allows us to prove that free products of polycyclic-by-finite groups amalgamated over finitely generated central subgroups are RFD. On the other hand we prove that the class of RFD C*-algebras (and groups) is not closed under central amalgamated free products. Namely we give an example of RFD groups (in fact finitely generated amenable RF groups) whose central amalgamated free product is not RFD, moreover it is not even maximally almost periodic. This answers a question of Khan and Morris [KM82].