On function and operator modules

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 $C^{\ast }$-algebra $C(K)$, a linear isometry $i\text{\hspace{0.17em}⁣}:X\to C(K)$, and a contractive unital homomorphism $\theta \text{\hspace{0.17em}⁣}:A\to C(K)$ such that $i(a\cdotp x) =\theta(a)i(x)$forany$a\in A, x\in X$. 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 $w^{\ast }$-version of the latter characterization, which generalizes some previous work of Effros and Ruan.