Hochschild cohomology of factors with property $\Gamma$

Abstract

The main result of this paper is that the $k^{\rm th}$ continuous Hochschild cohomology groups $H^k(\mathcal{M},\mathcal{M})$ and $H^k(\mathcal{M},B(H))$ of a von Neumann factor ${\mathcal{M}}\subseteq B(H)$ of type ${\rm II}_1$ with property $\Gamma$ are zero for all positive integers $k$. The method of proof involves the construction of hyperfinite subfactors with special properties and a new inequality of Grothendieck type for multilinear maps. We prove joint continuity in the $\|\cdot\|_2$-norm of separately ultraweakly continuous multilinear maps, and combine these results to reduce to the case of completely bounded cohomology which is already solved.