On the continuous cohomology of operator and Fréchet algabras
by
Zinaida A. Lykova
University of Newcastle, England

In the past ten years there has been an increasing interest in the computation of various types of continuous homologies and cohomologies of operator and Fréchet algebras. The homological theory now has applications in many branches of mathematics, including the spectral theory of commuting operators, K-theory and non-commutative differential geometry [1, 3].

We present some methods for the computation of the Hochschild and cyclic type continuous homologies and cohomologies. In particularly, we show relations between different types of homologies and cohomologies and we illustrate the use of cohomology relative to a subalgebra [4]. We also obtain the excision property for continuous cyclic and periodic cyclic homology and cohomology in the category of Fréchet algebras as a simple inference from the excision property for continuous Hochschild and bar homology [2]. Recall that an extension of Fréchet algebras 0 --> I --> A --> A/I --> 0 has the excision property in a particular homology H_* if there exists an associated long exact sequence

... --> Hn(I) --> Hn(A) --> Hn(A/I) --> Hn-1(I) --> ...

of the corresponding homology groups.

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A. Connes, Noncommutative geometry (Academic Press, London, 1994).

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J. Brodzki and Z.A. Lykova, `Excision in cyclic type homology of Fréchet algebras', to appear in Bull. London Math. Soc.

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J. Eschmeier and M. Putinar, Spectral Decompositions and Analytic Sheaves, London Math. Soc. Monographs, New Series 10 (Clarendon Press, Oxford, 1996).

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Z.A. Lykova, `Relative cohomology of Banach algebras', J. Operator Theory 41 (1999) 23-53.

Date received: June 1, 2000