Cyclic cohomology of certain nuclear Fréchet algebras and DF algebras

Zinaida Lykova

Open Mathematics (2008)

  • Volume: 6, Issue: 3, page 405-421
  • ISSN: 2391-5455

Abstract

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We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain ⊗ ^ -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 ⊗ ^ -algebras: the tensor algebra E ⊗ ^ 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.

How to cite

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Zinaida Lykova. "Cyclic cohomology of certain nuclear Fréchet algebras and DF algebras." Open Mathematics 6.3 (2008): 405-421. <http://eudml.org/doc/269245>.

@article{ZinaidaLykova2008,
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.},
author = {Zinaida Lykova},
journal = {Open Mathematics},
keywords = {Cyclic cohomology; Hochschild cohomology; nuclear DF-spaces; locally convex algebras; nuclear Fréchet algebra; cyclic cohomology},
language = {eng},
number = {3},
pages = {405-421},
title = {Cyclic cohomology of certain nuclear Fréchet algebras and DF algebras},
url = {http://eudml.org/doc/269245},
volume = {6},
year = {2008},
}

TY - JOUR
AU - Zinaida Lykova
TI - Cyclic cohomology of certain nuclear Fréchet algebras and DF algebras
JO - Open Mathematics
PY - 2008
VL - 6
IS - 3
SP - 405
EP - 421
AB - 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.
LA - eng
KW - Cyclic cohomology; Hochschild cohomology; nuclear DF-spaces; locally convex algebras; nuclear Fréchet algebra; cyclic cohomology
UR - http://eudml.org/doc/269245
ER -

References

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