On Limit Eigenvalue Distributions Associated to Residual Chains of Groups

TL;DR

This paper investigates the approximation of Brown measures of operators associated with residually finite groups, providing explicit examples where approximation fails and conditions where it succeeds.

Contribution

It presents an explicit example in the discrete Heisenberg group showing non-approximability of Brown measures and proves that finitely generated abelian groups allow such approximation.

Findings

Brown measure cannot always be approximated using finite quotients in residually finite groups.

In finitely generated abelian groups, the Brown measure can be approximated via finite quotients.

Explicit example in the discrete Heisenberg group demonstrating non-approximability.

Abstract

Let $G$ be a residually finite group. We give an explicit example in the discrete Heisenberg group that the Brown measure of multiplication operators $A \in \mathbb{Z}[G] \subseteq \mathcal{B}(\ell^2(G))$ in general can not be approximated using finite quotients $G/N$ of $G$. We show that in finitely generated abelian groups the Brown measure can be approximated using finite quotients.