On function and operator modules

Details On function and operator modules On function and operator modules

1999-06-14

arxiv.org/abs/math/9906099v1

Mathematics

Operator Algebras

Scientific paper

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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