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

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 $$k>0$$.

### 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
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