Bounded Derivations and the Hochschild Cohomology of Uniform Roe Algebras

Abstract

We first show that for a uniform Roe algebra associated to a bounded geometry metric space X, all bounded derivations from that uniform Roe algebra to itself are inner. We obtain this result using a “reduction of cocycles" method from Sinclair and Smith. Then the key technical ingredient comes from recent work of Braga and Farah in their paper “On the Rigidity of Uniform Roe Algebras".That all bounded derivations are inner is equivalent to the first norm continuous Hochschild cohomology group vanishing. It is then natural to ask if all the higher groups vanish. While we cannot answer this question completely, we are able to give necessary and sufficient conditions for the vanishing of the norm continuous Hochschild cohomology of a uniform Roe algebra. 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.