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Published online by Cambridge University Press:  02 February 2006

Zinaida A. Lykova
School of Mathematics and Statistics, University of Newcastle, Newcastle upon Tyne NE1 7RU, UK (
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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.

Research Article
Copyright © Edinburgh Mathematical Society 2006