Cyclic cohomology of certain nuclear Fréchet algebras and DF algebras

Abstract

We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain⊗-algebras. We use well-developed homological techniques together with some niceties of the theory of locally convex spaces to generalize the results known in the case of Banach algebras and their inverse limits to wider classes of topological algebras. To this end we show that, for a continuous morphism ϕ : X → Y of complexes of complete nuclear DF -spaces, the isomorphism of cohomology groups H n (ϕ) : H n (X ) → H n (Y) is automatically topological. The continuous cyclic-type homology and cohomology are described up to topological isomorphism for the following classes of biprojective⊗-algebras: the tensor algebra E⊗F generated by the duality (E, F, ·, · ) for nuclear Fréchet spaces E and F or for nuclear DF -spaces E and F ; nuclear biprojective Köthe algebras λ(P ) which are Fréchet spaces or DF -spaces; the algebra of distributions E * (G) on a compact Lie group G.

Amenable Köthe co‐echelon algebras

“…This class of objects has not been widely studied in the context of amenability. Although there are papers investigating homology/cohomology theory of DF‐algebras and also expositions treating general (in particular non‐Banach) topological algebras, e.g. Helemskii's monographs , this category is, in a sense, inconvenient to work with.…”

“…There is also no Open Mapping Theorem for general DF‐spaces. In order to be able to use this theorem Lykova and Taylor restrict their investigation to nuclear (in the sense of Grothendieck) DF‐spaces.…”

Hochschild cohomology of tensor products of topological algebras

“…The open mapping theorem holds in the categories of Fréchet spaces and of complete nuclear DF -spaces (see Corollary 3.2 for DF -spaces) and, for a continuous morphism of chain complexes ϕ : X → Y in these categories, a surjective map H n (ϕ) : H n (X ) → H n (Y) is automatically open (see [10,Lemma 0.5.9] and [17,Lemma 3.5]). …”

“…In [17,Theorem 5.4] we describe explicitly the cyclic-type homology and cohomology groups of amenable Fréchet algebras B. In particular, we show that all boundary maps of the standard homology complex C ∼ (B, B) have closed ranges.…”

“…Hochschild and cyclic cohomology of topological algebras play a prominent role in Ktheory [3] and in non-commutative geometry [2]. There are a number of papers addressing the calculation of Hochschild and cyclic continuous homology and cohomology of topological algebras (see, for example, [2,3,13,[15][16][17]23,25]). However, there are few explicit descriptions of non-trivial higher-dimensional Hochschild cohomology groups for topological algebras.…”