Relative cohomology of Banach algebras

Abstract

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.