The Hochschild Cohomology of Uniform Roe Algebras

TL;DR

This paper investigates the Hochschild cohomology of uniform Roe algebras, providing conditions for its vanishing and relating different types of Hochschild cohomology, extending previous results on derivations.

Contribution

It offers necessary and sufficient conditions for the vanishing of higher Hochschild cohomology groups of uniform Roe algebras and relates various cohomology notions.

Findings

Necessary and sufficient conditions for vanishing of $H^n_c(C_u^*(X),C_u^*(X))$

If norm continuous Hochschild cohomology vanishes, then ultraweak-weak* cohomology also vanishes

All bounded derivations on uniform Roe algebras are inner

Abstract

In Rufus Willett's and the authors paper "Bounded Derivations on Uniform Roe Algebras" we showed that all bounded derivations on a uniform Roe algebra $C^*_u(X)$ associated to a bounded geometry metric space $X$ are inner. This naturally leads to the question of whether or not the higher dimensional Hochschild cohomology groups of the uniform Roe algebra vanish also. While we cannot answer this question completely, we are able to give necessary and sufficient conditions for the vanishing of $H^n_c(C_u^*(X),C_u^*(X))$. Lastly, we show that if the norm continuous Hochschild cohomology of a uniform Roe algebra vanishes in all dimensions then the ultraweak-weak* continuous Hochschild cohomology of that uniform Roe algebra vanishes also.