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

Niels Grønbæk
2006 Transactions of the American Mathematical Society  
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 cohomology H n (A, A * ) vanishes for all n ≥ 1. The hypotheses of this theorem involve (i) strong H-unitality of A, (ii) a growth condition on diagonal matrices in A, and (iii) an extension of A in B by an amenable Banach algebra. As a corollary we show that if X is an
more » ... w 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 A is the Banach algebra of approximable operators on L p (X, µ, Ω) (1 ≤ p < ∞), then H n (A, A * ) = (0) for all n ≥ 0.
doi:10.1090/s0002-9947-06-03913-4 fatcat:3urzad6izjhntpix5kngbg3ohq