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arXiv · math/9906099

On function and operator modules

David P. Blecher·Christian Le Merdy
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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 $θ\colon A\to C(K)$ such that $i(a\cdotp x) =θ(a)i(x)$ 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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David P. Blecher, Christian Le Merdy. 1999-06-14. On function and operator modules. https://arxiv.org/abs/math/9906099

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