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Title: Splitting of the Hochschild cohomology of von Neumann algebras
Author: Drivaliaris, Dimosthenis
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 2000
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This thesis is concerned with the study of splitting for bounded and completely bounded Hochschild cohomology of von Neumann algebras. Having as a starting point the notions of a split and a split exact complex, which are standard in homological algebra, we define five types of splitting for the (completely) bounded Hochschild cohomology group of A, with coefficients in X, Hnc(b) (A,X). In general we could say that the study of splitting is the study of the invertibility of the coboundary map ∂n. We show that all types of splitting are closely connected to geometric properties of the space of n-boundaries Bn*(A, X) and of the space of n-cocycles Zn* (A,X) and discuss the relation between the different types of splitting. Then we define module actions on spaces of maps from, into and between A-modules. Given a A-module X and a space Y we make L1* (Y, X) into an A-module containing X. The modules L1*(Y, X) inherit duality and normality from the module X; the completely bounded case is particularly interesting since we have to define a matricial norm structure on the tensor product of two matricially normed spaces U and V such that the tracial dual of UV is completely isometrically isomorphic to the space of completely bounded maps from U into the tracial dual of V. On the other hand we define a module structure on L1* (X, Y) which generalises the notion of the dual A-module of X. The completely bounded case is again non-trivial because we must consider a new matricial norm structure on L1*(X, Y) generalising the matricial norm structure of the tracial dual. Duality and normality of L1* (X, Y) are also discussed. We continue by studying the relation between splitting and the modules L1* (Y, X).
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available