Abstract
We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain \( \hat \otimes \) -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 \( \hat \otimes \) -algebras: the tensor algebra E \( \hat \otimes \) 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 ε*(G) on a compact Lie group G.
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Lykova, Z.A. Cyclic cohomology of certain nuclear Fréchet algebras and DF algebras. centr.eur.j.math. 6, 405–421 (2008). https://doi.org/10.2478/s11533-008-0040-x
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DOI: https://doi.org/10.2478/s11533-008-0040-x