Title:

Splitting of the Hochschild cohomology of von Neumann algebras

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, H^{n}_{c(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 nboundaries B^{n}_{*}(A, X) and of the space of ncocycles Z^{n}_{* }(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 Amodules. Given a Amodule X and a space Y we make L^{1}_{*} (Y, X) into an Amodule containing X. The modules L^{1}_{*}(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 L^{1}_{*} (X, Y) which generalises the notion of the dual Amodule of X. The completely bounded case is again nontrivial because we must consider a new matricial norm structure on L^{1}_{*}(X, Y) generalising the matricial norm structure of the tracial dual. Duality and normality of L^{1}_{*} (X, Y) are also discussed. We continue by studying the relation between splitting and the modules L^{1}_{*} (Y, X).
