## Cyclic cohomology of projective limits of topological algebras

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### Abstract

We present methods for the computation of the Hochschild and cyclic continuous cohomology and homology of some locally convex topological algebras. Let (Aα,Tαβ) (Λ,≤) 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←αAα. We prove that, for the continuous cyclic cohomology HC* and continuous periodic cohomology HP* of A and Aα α ∈ Λ, for all n≥0, HCn(A) = lim→α HCn(A α), the inductive limit of HCn(Aα), and, for k = 0,1, HPk(A) = lim→α HP k(Aα). For a projective limit algebra A = lim ←m Am of a countable reduced projective system (Am, Tml)n of Fréchet algebras, we also establish relations between the cyclic-type continuous homology of A and A m, m∈ℕ. For example, we show the exactness of the following short sequence for all n≥0: equation presented 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 HC2l+k(A) starting from some integer l. We apply the above results to calculate the continuous cyclic-type homology and cohomology of some Fréchet locally m-convex algebras.

### Publication metadata

**Author(s): **Lykova ZA

**Publication type: **Article

**Publication status:** Published

**Journal: **Proceedings of the Edinburgh Mathematical Society

**Year: **2006

**Volume: **49

**Issue: **1

**Pages: **173-199

**Date deposited: **16/11/2012

**ISSN (print): **0013-0915

**ISSN (electronic): **1464-3839

**Publisher: **Cambridge University Press

**URL: **http://dx.doi.org/10.1017/S0013091504000410

**DOI: **10.1017/S0013091504000410

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