Introduction

This chapter presents some applications of cohomology to the structure theory of von Neumann algebras. Much of this material applies to arbitrary Banach algebras without modification, and so we have stated many of the results in full generality.

Section 7.2 explores the relationship between the first cohomology group and the principal component in the automorphism group of a Banach algebra. The main result (7.2.5) shows that, under the hypothesis H1(A, A) = 0, any automorphism in the principal component of Aut(A) is not only inner but is implemented by an element of the principal component of the group A−1 of invertible elements in A.

Section 7.3 contains an implicit function theorem for Fréchet differentiable maps on Banach spaces, which we apply to questions of stability in Section 7.4. Here we present two main results. If λ: A → A is an invertible linear map then (x, y) → λ−1(λ(x)λ(y)) defines a new associative product on A. Under the hypothesis of vanishing second and third cohomology groups, we show that any associative product sufficiently close to the original one has such a form, and λ can be chosen to be close to the identity map. The second stability result is a consequence of this: any von Neumann algebra sufficiently close to an injective von Neumann algebra must itself be injective, and the two algebras are isomorphic.