Hochschild cohomology of type II$_1$ von Neumann algebras with Property $\Gamma$

TL;DR

This paper introduces a generalized Property Γ for type II$_1$ von Neumann algebras and proves that such algebras have vanishing higher Hochschild cohomology groups, extending previous results.

Contribution

It generalizes Property Γ to a broader class of type II$_1$ von Neumann algebras and establishes the vanishing of their higher Hochschild cohomology groups.

Findings

Property Γ is extended to type II$_1$ von Neumann algebras.

Higher Hochschild cohomology groups vanish for these algebras.

Generalizes earlier results by Christensen et al.

Abstract

In this paper, Property $\Gamma$ for a type II$_{1}$ von Neumann algebra is introduced as a generalization of Murray and von Neumann's Property $\Gamma$ for a type II$_{1}$ factor. The main result of this paper is that if a type II$_{1}$ von Neumann algebra $\mathcal{M}$ with separable predual has Property $\Gamma$, then the continuous Hochschild cohomology group $H^{k}(\mathcal{M}, \mathcal{M})$ vanishes for every $k \geq 2$. This gives a generalization of an earlier result due to E. Christensen, F. Pop, A.M. Sinclair and R.R. Smith.