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

الأصل · EN

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 Hⁿ(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, L₁(μ,Ω) is an infinite dimensional L₁-space, and A is the Banach algebra of approximable operators on Lₚ(X,μ,Ω), (1=<p<∞), then Hⁿ(A,A*)=(0) for all n>=0.