The Hochschild cohomology problem for von Neumann algebras

In 1967, when Kadison and Ringrose began the development of continuous cohomology theory for operator algebras, they conjectured that the cohomology groups H(n)(M, M), n ≥ 1, for a von Neumann algebra M, should all be zero. This conjecture, which has important structural implications for von Neumann algebras, has been solved affirmatively in the type I, II(∞), and III cases, leaving open only the type II(1) case. In this paper, we describe a positive solution when M is type II(1) and has a Cartan subalgebra and a separable predual.