Smooth cohomology of $C^{\ast }$-algebras

Abstract

We define a notion of smooth cohomology for $C^{\ast }$-algebras which admit a faithful trace. We show that if $ \A\subseteq B(\h) $ is a $C^{\ast }$-algebra with a faithful normal trace $\tau$ on the ultra-weak closure $ \bar{\A} $ of $\mathcal{A}$, and $X$ is a normal dual operatorial $ \bar{\A}$-bimodule, then the first smooth cohomology ${\mathcal{H}}_s^1(\mathcal{A},X)$ of $\mathcal{A}$ is equal to ${\mathcal{H}}^1(\mathcal{A},X_{\tau })$, where $X_{\tau }$ is a closed submodule of $X$ consisting of smooth elements.