General Introduction

The theory of multilinear maps on a von Neumann algebra is developed in these notes and applied to the continuous Hochschild cohomology of von Neumann algebras. The methods used are those of von Neumann algebras and complete boundedness rather than of homological algebra, and only elementary cohomlogical techniques are employed in the proofs. We have chosen to base our presentation on the problem of whether the continuous cohomology groups Hn(M, M) of a von Neumann algebra M over itself are zero for all n. This, and closely related questions, has stimulated much of the recent development of the theory of completely bounded maps, and so we have adopted an approach which has wider applications beyond cohomology theory. The results in these notes have been proved in full generality, provided that they do not stray too far from the central topic of dual normal modules over von Neumann algebras.

There are two main reasons for investigating the Hochschild cohomology groups of operator algebras. When they are non-zero they provide invariants which can distinguish classes of algebras; when they are zero they lead to positive results on the stability of algebraic structures and on the space of bounded derivations on an operator algebra. Elliott's classification of separable AF-C* -algebras by K-theory is an example of the use of a homological invariant [El1].