Vanishing of H2w(m, K(h)) for Certain Finite Von Neumann Algebras
نویسندگان
چکیده
The cohomology of operator algebras introduced by B. E. Johnson, R. V. Kadison, and J. R. Ringrose in a series of three papers is a useful tool for obtaining new invariants for operator algebras or to prove stability results by the vanishing of their cohomology groups (see [14]). If X is a von Neumann algebra and a Banach bimodule over M, and if « is a positive integer, then the nth cohomology group of M is denoted by H"(M, X). If X is also a normal dual bimodule, then H"W(M, X) is the «th weakly continuous cohomology group and it is proved in [14] that H"(M, X) is isomorphic to H^(M, X) and vanishes whenever M is approximately finite dimensional. Actually, the results of Alain Connes show that the vanishing of Hx (M, X) for every normal dual Banach bimodule over M is equivalent to the injectivity of M (see [2]). If M c B(H) (the space of all linear bounded operators on a Hubert space H ), then B(H) itself is a normal dual Banach bimodule over M ; the most interesting examples of dual normal bimodules are B(H) and M. By the work of E. Christensen (see [3]) it is known that Hx (M, B(H)) = 0 in most cases (see also [5] for results concerning the higher cohomology groups). It was very well known that Hc (M, M) for nonnegative M vanishes foxk=\, but for k = 2 nothing was known (excepting an example of B. E. Johnson [9]) until recently. It was proved by E. Christensen and A. Sinclair [6] that it vanishes if M has property Y. When X is no longer a normal dual bimodule, the proof of vanishing of H"(M, X) can be difficult even for injective M (see [9]).
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