Smooth cohomology of $ C^* $-algebras

TL;DR

This paper introduces a new notion of smooth cohomology for $ C^* $-algebras with faithful traces and establishes an equivalence between smooth and standard cohomology under certain conditions involving dual operator bimodules.

Contribution

It defines smooth cohomology for $ C^* $-algebras with traces and proves an isomorphism between smooth and standard first cohomology groups for specific bimodules.

Findings

Smooth cohomology is well-defined for $ C^* $-algebras with faithful traces.

First smooth cohomology coincides with standard cohomology for certain bimodules.

The result applies to $ C^* $-algebras with ultra-weakly closed subalgebras and normal dual bimodules.

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.