Hochschild cohomology of factors with property \(\Gamma\). (English) Zbl 1048.46051

Christensen, Erik; Pop, Florin; Sinclair, Allan M.; Smith, Roger R.

Ann. Math. (2) 158, No. 2, 635-659 (2003).

Let \({\mathcal M}\subset B(H)\) be a type \(\text{II}_1\) factor with property \(\Gamma\). It is proved that the continuous Hochschild cohomology groups \(H^k({\mathcal M},{\mathcal M})\) and \(H^k({\mathcal M},B(H))\) vanish for all \(k>0\). The method of proof involves the construction of hyperfinite subfactors with special properties and a new inequality of Grothendieck type for multilinear maps, which allows to prove joint continuity, with respect to the \(\| \cdot\| _2\)-norm, of separately ultraweakly continuous multilinear maps. This implies that \(H^k({\mathcal M},{\mathcal X})=H^k_{\text{cb}}({\mathcal M},{\mathcal X})\), \(k>0\), for any ultraweakly closed \({\mathcal M}\)-bimodule \({\mathcal X}\) lying between \({\mathcal M}\) and \(B(H)\), where \(H^k_{\text{cb}}\) denotes completely bounded cohomology. The main result follows from the vanishing of \(H^k_{\text{cb}}({\mathcal M},{\mathcal X})\) for \({\mathcal X}\) being either \({\mathcal M}\) or \(B(H)\) and for \(k>0\).

Reviewer: V. M. Manuilov (Moskva)

MSC:

46L37	Subfactors and their classification
46L10	General theory of von Neumann algebras
46M18	Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.)
19D55	\(K\)-theory and homology; cyclic homology and cohomology

Keywords:

continuous Hochschild cohomology; completely bounded cohomology