  We give a solution, via operator spaces, of an old problem in the Morita
equivalence of C*-algebras. Namely, we show that C*-algebras are strongly
Morita equivalent in the sense of Rieffel if and only if their categories of
left operator modules are isomorphic via completely contractive functors.
Moreover, any such functor is completely isometrically isomorphic to the
Haagerup tensor product (= interior tensor product) with a strong Morita
equivalence bimodule. An operator module over a C$^*-$algebra $\A$ is a closed
subspace of some B(H) which is left invariant under multiplication by
$\pi(\A)$, where $\pi$ is a *-representation of $\A$ on $H$. The category
$_{\A}HMOD$ of *-representations of $\A$ on Hilbert space is a full subcategory
of the category $_{\A}OMOD$ of operator modules. Our main result remains true
with respect to subcategories of $OMOD$ which contain $HMOD$ and the
C$^*-$algebra itself. It does not seem possible to remove the operator space
framework; in the very simplest cases there may exist no bounded equivalence
functors on categories with bounded module maps as morphisms (as opposed to
completely bounded ones). Our proof involves operator space techniques,
together with a C$^*-$algebra argument using compactness of the quasistate
space of a C$^*-$algebra, and lowersemicontinuity in the enveloping von Neumann
algebra.
