Cyclic cohomology of certain nuclear Fr\'echet and DF algebras

TL;DR

This paper provides explicit formulas for continuous Hochschild and cyclic homology and cohomology of specific nuclear topological algebras, demonstrating topological isomorphisms and describing these invariants for various classes of biprojective algebras.

Contribution

It establishes that cohomology isomorphisms are automatically topological for complexes of nuclear $DF$-spaces and describes cyclic homology for several classes of biprojective $ ext{ extasciitilde} ext{-algebras}.

Findings

Cohomology isomorphisms are topological for complexes of nuclear $DF$-spaces.

Explicit formulas for cyclic homology of tensor and K"othe algebras.

Topological cyclic homology described for distributions on compact Lie groups.

Abstract

We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain topological algebras. To this end we show that, for a continuous morphism $\phi: \X\to \Y$ of complexes of complete nuclear $DF$-spaces, the isomorphism of cohomology groups $H^n(\phi): H^n(\X) \to H^n(\Y)$ is automatically topological. The continuous cyclic-type homology and cohomology are described up to topological isomorphism for the following classes of biprojective $\hat{\otimes}$-algebras: the tensor algebra $E \hat{\otimes} F$ generated by the duality $(E, F, < \cdot, \cdot >)$ for nuclear Fr\'echet spaces $E$ and $F$ or for nuclear $DF$-spaces $E$ and $F$; nuclear biprojective K\"{o}the algebras $\lambda(P)$ which are Fr\'echet spaces or $DF$-spaces; the algebra of distributions $\mathcal{E}^*(G)$ on a compact Lie group $G$.