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

Author: Niels Grønbæk
Journal: Trans. Amer. Math. Soc. 358 (2006), 2651-2662
MSC (2000): Primary 46M20; Secondary 47B07, 16E40
Published electronically: January 24, 2006
MathSciNet review: 2204050
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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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Additional Information

Niels Grønbæk
Affiliation: Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark

Keywords: Bounded Hochschild cohomology, H-unital, simplicially trivial
Received by editor(s): December 2, 2003
Received by editor(s) in revised form: August 3, 2004
Published electronically: January 24, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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