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

Grønbæk, Niels

doi:10.1090/S0002-9947-06-03913-4

Bounded Hochschild cohomology of Banach algebras with a matrix-like structure
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Trans. Amer. Math. Soc. 358 (2006), 2651-2662 Request permission

Abstract:

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 $n\geq 0$.

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