On Hochschild homology of uniform Roe algebras with coefficients in uniform Roe bimodules

TL;DR

This paper investigates the Hochschild homology of uniform Roe algebras with coefficients in bimodules, extending understanding of derivations and cohomology in the context of metric spaces with bounded geometry.

Contribution

It computes the space of outer derivations and explores higher Hochschild cohomology for uniform Roe algebras with various bimodule coefficients.

Findings

Outer derivations are characterized for uniform Roe algebras with specific bimodules.

Higher Hochschild cohomology groups are computed or bounded.

Results depend on the metric properties of the underlying space.

Abstract

It was shown recently by M. Lorentz and R. Willett that all bounded derivations of the uniform Roe algebras of metric spaces of bounded geometry are inner. Here we calculate the space of outer derivations of the uniform Roe algebras with coefficients in uniform Roe bimodules related to various metrics on the two copies of the given space. We also give some results on the higher Hochschild cohomology with coefficients in uniform Roe bimodules.