TAILIEUCHUNG - Đề tài " Hochschild cohomology of factors with property Γ "

Dedicated to the memory of Barry Johnson, 1937–2002 Abstract The main result of this paper is that the k th continuous Hochschild cohomology groups H k (M, M) and H k (M, B(H)) of a von Neumann factor M ⊆ B(H) of type II1 with property Γ 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 · 2 -norm of separately ultraweakly continuous multilinear maps, and combine these results to reduce to. | Annals of Mathematics Hochschild cohomology of factors with property r By Erik Christensen Florin Pop Allan M. Sinclair and Roger R. Smith - Annals of Mathematics 158 2003 635 659 Hochschild cohomology of factors with property r By Erik Christensen Florin Pop Allan M. Sinclair and Roger R. Smith - Dedicated to the memory of Barry Johnson 1937-2002 Abstract The main result of this paper is that the kth continuous Hochschild cohomology groups Hk M M and Hk M B H of a von Neumann factor M c B H of type III with property r 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 II 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. 1. Introduction The continuous Hochschild cohomology of von Neumann algebras was initiated by Johnson Kadison and Ringrose in a series of papers 21 23 24 where they developed the basic theorems and techniques of the subject. From their results and from those of subsequent authors it was natural to conjecture that the kth continuous Hochschild cohomology group Hk M M of a von Neumann algebra over itself is zero for all positive integers k . This was verified by Johnson Kadison and Ringrose 21 for all hyperfinite von Neumann algebras and the cohomology was shown to split over the center. A technical version of their result has been used in all subsequent proofs and is applied below. Triviality of the cohomology groups has interesting structural implications for von Neumann algebras 39 Chapter 7 which surveys the original work in this area by Johnson 20 and Raeburn and Taylor 35 and so it is important to determine when this occurs. Partially supported by a Scheme 4 collaborative grant from the London Mathematical Society. t Partially supported by a grant from

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