On function and operator modules

Blecher, David; Le Merdy, Christian

doi:10.1090/S0002-9939-00-05866-4

On function and operator modules
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by David Blecher and Christian Le Merdy

Proc. Amer. Math. Soc. 129 (2001), 833-844

DOI: https://doi.org/10.1090/S0002-9939-00-05866-4

Published electronically: August 30, 2000

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Abstract:

Let $A$ be a unital Banach algebra. We give a characterization of the left Banach $A$-modules $X$ for which there exists a commutative unital $C^{*}$-algebra $C(K)$, a linear isometry $i\colon X\to C(K)$, and a contractive unital homomorphism $\theta \colon A\to C(K)$ such that $i(a\cdotp x) =\theta (a)i(x)$ for any $a\in A,\ x\in X$. We then deduce a “commutative" version of the Christensen-Effros-Sinclair characterization of operator bimodules. In the last section of the paper, we prove a $w^{*}$-version of the latter characterization, which generalizes some previous work of Effros and Ruan.

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