A generalization of expander graphs and local reflexivity of uniform Roe algebras

TL;DR

This paper introduces weak expander sequences and demonstrates that their uniform Roe algebras lack local reflexivity, extending the understanding of the algebraic properties of expanders and their generalizations.

Contribution

It generalizes expander graphs to weak expander sequences and analyzes their uniform Roe algebras within a unified framework using generalized box spaces.

Findings

Uniform Roe algebras of weak expander sequences are not locally reflexive.

Uniform Roe algebras of expander graphs are not exact.

Introduces generalized box spaces to unify the study of box spaces and expander sequences.

Abstract

We introduce a generalization of expander graphs, which is called a weak expander sequence. It is proved that a uniform Roe algebra of a weak expander sequence is not locally reflexive. It follows that uniform Roe algebras of expander graphs are not exact. We introduce the notion of a generalized box space to discuss box spaces and expander sequences in a unified framework. Key tools for the proof are amenable traces and measured groupoids associated to generalized box spaces.