The Hochschild cohomology problem for von Neumann algebras

摘要

In 1967, when Kadison and Ringrose began the development of continuous cohomology theory for operator algebras, they conjectured that the cohomology groups Hⁿ(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 II1 case. In this paper, we describe a positive solution when M is type II1 and has a Cartan subalgebra and a separable predual.