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Cyclic-type cohomology of strict inductive limits of Fréchet algebras

Lookup NU author(s): Dr Zinaida LykovaORCiD


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We present methods for the computation of the Hochschild and cyclic-type continuous homology and cohomology of some locally convex strict inductive limits A = lim→m Am of Fréchet algebras Am. In the pure algebraic case it is known that, for the cyclic homology of A, HCn = lim→m HCn(Am) for all n ≥ 0 [Cyclic Homology, Springer, Berlin, 1992, E.2.1.1]. We show that, for a locally convex strict inductive system of Fréchet algebras (Am)m=1∞ such that 0 → Am → Am+1 → Am+1/Am → 0 is topologically pure for each m and for continuous Hochschild and cyclic homology, similar formulas hold. For such strict inductive systems of Fréchet algebras we also establish relations between the continuous cohomology of A and Am, m ε N. For example, for the continuous cyclic cohomology ℋCn and ℋCn(Am), m ε N, we show the exactness of the following short sequence, for all n ≥ 0, 0 → limm←(1)ℋCn-1 (Am) → ℋCn(A) → limi← ℋCn (Am) → 0, where limm←(1) is the first derived functor of the projective limit. We give explicit descriptions of continuous periodic and cyclic homology and cohomology of a LF-algebra A = limm→Am which is a locally convex strict inductive limit of amenable Banach algebras Am, where for each m, Am is a closed ideal of Am+1. © 2005 Elsevier B.V. All rights reserved.

Publication metadata

Author(s): Lykova ZA

Publication type: Article

Publication status: Published

Journal: Journal of Pure and Applied Algebra

Year: 2006

Volume: 205

Issue: 3

Pages: 471-497

Date deposited: 28/04/2011

ISSN (print): 0022-4049

ISSN (electronic): 1873-1376

Publisher: Elsevier BV


DOI: 10.1016/j.jpaa.2005.07.014


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