Reactive cohomology of Banach algebras

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 HCn(A) of A and the Banach B-relative cyclic cohomology group HCBn(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, HCn(A) = HCBn(A) for all n greater than or equal to 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.