Journal of Operator Theory

Volume 41, Issue 1, Winter 1999 pp. 23-53.

Relative cohomology of Banach algebras

Authors: Zinaida A. Lykova
Author institution: Department of Mathematics, Merz Court, University of Newcastle, Newcastle upon Tyne NE1 7RU, England

Summary: Let

$A$ be a Banach algebra, not necessarily unital, and let

$B$ be a closed subalgebra of

$A$ . We establish a connection between the Banach cyclic cohomology group

${\cal HC}^n(A)$ of

$A$ and the Banach

$B$ -relative cyclic cohomology group

${\cal HC}^n_B(A)$ of

$A$ . We prove that, for a Banach algebra

$A$ with a bounded approximate identity and an amenable closed subalgebra

$B$ of

$A$ , up to topological isomorphism,

${\cal HC}^n(A) = {\cal HC}^n_B(A)$ for all

$n \ge 0$ . We also establish a connection between the Banach simplicial or cyclic cohomology groups of

$A$ and those of the quotient algebra

$A/I$ by an amenable closed bi-ideal

$I$ . The results are applied to the calculation of these groups for certain operator algebras, including von Neumann algebras and joins of operator algebras.

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