The Hochschild Cohomology of Uniform Roe Algebras

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.