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

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