Hochschild Cohomology of Von Neumann AlgebrasThe subject of this book is the continuous Hochschild cohomology of dual normal modules over a von Neumann algebra. The authors develop the necessary technical results, assuming a familiarity with basic C*-algebra and von Neumann algebra theory, including the decomposition into types, but no prior knowledge of cohomology theory is required and the theory of completely bounded and multilinear operators is given fully. Those cases when the continuous Hochschild cohomology Hn(M,M) of the von Neumann algebra M over itself is zero are central to this book. The material in this book lies in the area common to Banach algebras, operator algebras and homological algebra, and will be of interest to researchers from these fields. |
Common terms and phrases
A₁ amenable An+1 automorphism averaging B-module map Banach algebras Banach space bounded linear bounded operators C*-algebra C*-subalgebra Cartan subalgebra coboundary cocycle cohomology groups completely bounded maps contains continuous linear map Corollary defined definition denote derivation dual normal M-module e₁ element equation exists finite dimensional subalgebras follows Funct Grothendieck inequality Haagerup tensor product Hence Hilbert space Hochschild cohomology hyperfinite hyperfinite von Neumann II₁ von Neumann implies injective isomorphic l¹(G L²(M Lemma linear span M₂ mann algebra masa minimal projections Mn(V module maps multilinear maps n-linear Neumann algebra Neumann subalgebra Note operator algebras operator spaces orthogonal predual Proof prove representation Ringrose self-adjoint separable Hilbert space separable predual sequence space H subspace tensor product Theorem theory topology type II₁ factor ultraweak closure ultraweakly continuous ultraweakly dense vector von Neumann algebra zero