Uniform Roe coronas

TL;DR

This paper explores the relationship between uniform Roe coronas and the coarse geometry of underlying metric spaces, showing consistency results and independence from ZFC under certain conditions.

Contribution

It establishes conditions under which isomorphism of uniform Roe coronas implies coarse equivalence or bijective coarse equivalence, and demonstrates independence results from ZFC.

Findings

Isomorphism implies coarse equivalence under certain set-theoretic assumptions.

For spaces with property A, isomorphism corresponds to bijective coarse equivalence on cofinite subsets.

Some uniform Roe coronas' isomorphism is independent of ZFC.

Abstract

A uniform Roe corona is the quotient of the uniform Roe algebra of a metric space by the ideal of compact operators. Among other results, we show that it is consistent with ZFC that isomorphism between uniform Roe coronas implies coarse equivalence between the underlying spaces, for the class of uniformly locally finite metric spaces which coarsely embed into a Hilbert space. Moreover, for uniformly locally finite metric spaces with property A, it is consistent with ZFC that isomorphism between the uniform Roe coronas is equivalent to bijective coarse equivalence between some of their cofinite subsets. We also find locally finite metric spaces such that the isomorphism of their uniform Roe coronas is independent of ZFC. All set-theoretic considerations in this paper are relegated to two 'black box' principles.