Smooth cohomology of $ C^* $-algebras

Abstract: 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.