CYCLIC COHOMOLOGY OF PROJECTIVE LIMITS OF TOPOLOGICAL ALGEBRAS

Abstract

We present methods for the computation of the Hochschild and cyclic continuous cohomology and homology of some locally convex topological algebras. Let $(A_{\alpha},T_{\alpha,\beta})_{(\varLambda,\le)}$ be a reduced projective system of complete Hausdorff locally convex algebras with jointly continuous multiplications, and let $A$ be the projective limit algebra $A=\lim_{\substack{\raisebox{-3pt}{\tiny$\leftarrow$}\\ \raisebox{2pt}{\tiny$\,\alpha$}}} A_\alpha$. We prove that, for the continuous cyclic cohomology $HC^*$ and continuous periodic cohomology $HP^*$ of $A$ and $A_\alpha$, $\alpha\in\varLambda$, for all $n\ge0$, $HC^n(A)=\lim_{\substack{\raisebox{-3pt}{\tiny$\rightarrow$}\\ \raisebox{2pt}{\tiny$\!\alpha\,$}}} HC^n(A_\alpha)$, the inductive limit of $HC^n(A_\alpha)$, and, for $k=0,1$, $HP^k(A)=\lim_{\substack{\raisebox{-3pt}{\tiny$\rightarrow$}\\ \raisebox{2pt}{\tiny$\!\alpha\,$}}} HP^k(A_\alpha)$. For a projective limit algebra $A=\lim_{\substack{\raisebox{-3pt}{\tiny$\leftarrow$}\\ \raisebox{2pt}{\tiny$\,m$}}}A_m$ of a countable reduced projective system $(A_m,T_{m,\ell})_{\mathbb{N}}$ of Fréchet algebras, we also establish relations between the cyclic-type continuous homology of $A$ and $A_m$, $m\in\mathbb{N}$. For example, we show the exactness of the following short sequence for all $n\ge0$:

$$ 0\rightarrow {\lim_{\substack{\leftarrow\\ \raisebox{2pt}{\scriptsize$\,m$}}}}^{1}HC_{n+1}(A_m)\rightarrow HC_n(A)\rightarrow \lim_{\substack{\leftarrow\\ \raisebox{2pt}{\scriptsize$\,m$}}} HC_{n}(A_m)\rightarrow0. $$

We present a class of Fréchet algebras $A$ for which the continuous periodic cohomology $HP^k(A)$, $k=0,1$, is isomorphic to the continuous cyclic cohomology $HC^{2\ell+k}(A)$ starting from some integer $\ell$. We apply the above results to calculate the continuous cyclic-type homology and cohomology of some Fréchet locally $m$-convex algebras.