A Cohomological Characterization of Approximately Finite Dimensional Von Neumann Algebras

One of the purposes in the computation of cohomology groups is to establish invariants which may be helpful in the classification of the objects under consideration. In the theory of continuous Hochschild cohomology for operator algebras R. V. Kadison and J. R. Ringrose proved [10] that for any hyperfinite von Neumann algebra M and any dual normal M-bimodule S, all the continuous cohomology groups vanish. Based on his fundamental paper [4] on injective von Neumann algebras A. Connes established in 1978 in [5] a converse statement to the result by Kadison and Ringrose. Thus we have a characterization of approximately finite dimensionality for von Neumann algebras by the vanishing of all continuous Hochschild cohomology. When examining the proofs in [1, 5] one finds that there is a single module and a single derivation which has the characterizing property such that M is an injective von Neumann algebra if and only if this test derivation is inner. The equivalence between injectivity and approximately finite dimensionality was first established in [4]. A later proof can be found in [7] and [6] contains an extension to algebras on non separable Hilbert spaces. Having this we can state that a von Neumann algebra is approximately finite dimensional if and only if Connes’ test derivation or test 1-cocycle is a coboundary. Various researchers have asked whether the vanishing of every continuous second cohomology group for a von Neumann algebra M with coefficients in a dual normal module would imply that M is approximately finite dimensional. We can not settle this question here, but we are able to construct a Banach M-bimodule S and a continuous 2-cocycle Φ on M with coefficients in S such that M is approximately finite dimensional if and only if Φ cobounds. This is the only result in this paper. The module S and the 2-cocycle Φ is constructed as a “2-dimensional” straightforward generalization of the module and the test derivation from [5]. In [5] the derivation δ : M −→ Y ⊆ B(B(H)) is given by δ(m)(x) = [m,x]. The exact description of the M-module Y is left out here, but Y is clearly a dual normal M-bimodule. The 2-cocycle Φ we use in this example is constructed analogously by