Smooth cohomology of $ C^* $-algebras

We define a notion of smooth cohomology for $C^*$-algebras which admit a faithful trace. We show that if $\A\subseteq B(\h)$ is a $C^*$-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 $\mathcal{H}^1_{s}(\mathcal{A},X)$ 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.