Geodesic
Hochschild cohomology of factors with property $\Gamma$
Erik Christensen1 ; Florin Pop2 ; Allan M. Sinclair3 ; Roger R. Smith4
1 Institute for Mathematiske FAG, University of Copenhagen, 2100 Copenhagen, Denmark
2 Department of Mathematics, Wagner College, Staten Island, NY 10301, United States
3 The School of Mathematics, The University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
4 Department of Mathematics, Texas A & M University, College Station, TX 77843, United States
Annals of mathematics, Tome 158 (2003) no. 2, pp. 635-659

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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.

DOI : 10.4007/annals.2003.158.635

Erik Christensen 1 ; Florin Pop 2 ; Allan M. Sinclair 3 ; Roger R. Smith 4

1 Institute for Mathematiske FAG, University of Copenhagen, 2100 Copenhagen, Denmark
2 Department of Mathematics, Wagner College, Staten Island, NY 10301, United States
3 The School of Mathematics, The University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
4 Department of Mathematics, Texas A & M University, College Station, TX 77843, United States 
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Erik Christensen; Florin Pop; Allan M. Sinclair; Roger R. Smith. Hochschild cohomology of factors with property $\Gamma$. Annals of mathematics, Tome 158 (2003) no. 2, pp. 635-659. doi: 10.4007/annals.2003.158.635

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