The cyclic homology of crossed product algebras, II. Topological algebras.

Abstract

This article is a sequel to [6], in which we constructed a spectral sequence for the cyclic homology of a crossed product algebra A o G, where A is an algebra and G is a discrete group acting on A. In this article, we will show how similar results hold when A is a topological algebra, and G is a Lie group acting differentiably on A. We assume the notation and results of [6], which we will refer to as Part I. All of the topological vector spaces considered in this article will be locally convex, complete and Hausdorff. A topological algebra is a topological vector space A with an associative product m : A×A −→ A which is separately continuous. This definition motivates the introduction of Grothendieck’s inductive tensor product V1⊗V2 of two topological vector spaces, which is the completion of the algebraic tensor product V1⊗V2 with respect to the finest compatible tensor product topology, in the sense of Grothendieck ([7], page 89). Recall some of the properties of the inductive tensor product: