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.

Authors

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