Hochschild cohomology of factors with property $\Gamma$


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.


Erik Christensen

Institute for Mathematiske FAG, University of Copenhagen, 2100 Copenhagen, Denmark

Florin Pop

Department of Mathematics, Wagner College, Staten Island, NY 10301, United States

Allan M. Sinclair

The School of Mathematics, The University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom

Roger R. Smith

Department of Mathematics, Texas A & M University, College Station, TX 77843, United States