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

Abstract

In this paper, Property $Γ$ for a type II$_{1}$ von Neumann algebra is introduced as a generalization of Murray and von Neumann's Property $Γ$ 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 $Γ$, 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.