On Morita's fundamental theorem for $C^*$-algebras

Abstract

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 $\mathcal A$ is a closed subspace of some B(H) which is left invariant under multiplication by $\pi(\mathcal\ A)$, where $\pi$ is a*-representation of $\mathcal A$ on $H$. The category $_{\mathcal{AHMOD}}$ of *-representations of $\mathcal A$ on Hilbert space is a full subcategory of the category $_{\mathcal{AOMOD}}$ 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.