Bounded Hochschild cohomology of Banach algebras with a matrix-like structure

Abstract

Let B be a unital Banach algebra. A projection in B which is equivalent to the identitity may give rise to a matrix-like structure on any two-sided ideal A in B. In this set-up we prove a theorem to the effect that the bounded Hochschild cohomology H^n(A,A^*) vanishes for all n>=1. The hypothesis of this theorem involve (i) strong H-unitality of A, (ii) a growth condition on diagonal matrices in A, (iii) an extension of A in B with trivial bounded simplicial homology. As a corollary we show that if X is an infinite dimensional Banach space with the bounded approximation property, L_1(\mu,\Omega) is an infinite dimensional L_1-space, and A is the Banach algebra of approximable operators on L_p(X,\mu,\Omega), (1=<p<\infty), then H^n(A,A^*)=(0) for all n>=0.