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

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 $k\ge 2$. This gives a generalization of an earlier result due to E. Christensen, F. Pop, A.M. Sinclair and R.R. Smith.