# HOCHSCHILD COHOMOLOGY OF TYPE Pi(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 $k \geq 2$. This gives a generalization of an earlier result due to E. Christensen, F. Pop, A.M. Sinclair and R.R. Smith.

• Qian, Wenhua
• Shen, Junhao

### Publication Date

• September 2015

### Keywords

• Hochschild cohomology
• property Gamma
• von Neumann algebra

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