I'm trying to get an understanding of Hilbert $C^*$-bi-modules from a geometric point of view. As is well-known, we have that
i) Commutative unital $C^*$-algebras correspond to compact Hausdorff spaces through the Gelfand--Naimark theorem and the identification $X \mapsto C(X)$.
ii) In this setting, Hermtian vector bundles are generalized as Hilbert $C^*$-modules.
Does the notion of a Hilbert $C^*$-bimodule, over a pair of $C^*$-algebras $(A,B)$, have a corresponding geometric interpretation in the classical case?
My guess, since we are now dealing with two $C^*$-algebras, there must a second space whose functions can multiply the sections of the bundle. The only way I can think of doing this is if the new space is simply a subspace of the first. But this doesn't feel like the "correct picture" here.