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

Abstract

Let '8 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 in. In this set-up we prove a theorem to the effect that the bounded cohomology vanishes for all n > 1. The hypotheses of this theorem involve (i) strong H-unitality of, (ii) a growth condition on diagonal matrices in, and (iii) an extension of in by an amenable Banach algebra. As a corollary we show that if X is an infinite dimensional Banach space with the bounded approximation property, L 1 (μ, Ω) is an infinite dimensional L 1 -space, and 21 is the Banach algebra of approximable operators on Lp(X, μ,Ω) (1 ≤p ≤ ∞), then = (0) for all n ≥ 0.