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Smooth bimodules and cohomology of II_1 factors


The work in this thesis, which is based to a large extent on a joint paper with Professor

Popa [GP14], provides a systematic study of derivations of von Neumann algebras M with values in Banach M -bimodules that satisfy a combination of smoothness and operatorial conditions. A typical example of such a smooth bimodule for M is the ideal of compact operators K(H) on the Hilbert space H on which M acts, a case that was fully analyzed in [JP72; Pop84], where it was shown that any derivation of M into K(H) is inner, i.e. it is implemented by an element in K(H).

However, we show that the class of smooth subbimodules of B(H) is much larger. We then provide a far reaching generalizations of the results in [JP72, Pop84], by showing that given any norm-closed smooth M -subbimodule B0 B(H), any derivation of M into B0

is inner. Moreover, we prove that if the von Neumann algebra M is either properly infnite, or it is finite but has \good decomposability" properties (e.g, if it contains a diuse amenable wq-normal von Neumann subalgebra), then given any abstract Banach M -bimodule B_0 satisfying the operatorial and smoothness conditions, all derivations of M into B0 are inner.

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