Unit Vectors, Morita Equivalence and Endomorphisms

With every (strict or normal) unital endomorphism of the algebra of all adjointable operators on a Hilbert module there is associated a correspondence (that is, a Hilbert bimodule) such that the endomorphism can be recovered as amplification of the identity representation with that correspondence. In these notes we show the converse of this statement in the case of strongly full W–correspondences by establishing that every W–correspondence is Morita equivalent to one that has a unit vector. This, actually, means that every discrete product system of strongly full W–correspondences comes from a discrete E0–semigroup. (We also show the C–analogue of this result in the special case of full C–correspondences over a unital C–algebras.) Taking into account the duality between a von Neumann correspondence (that is, a W–correspondence over a von Neumann algebra) and its commutant, we furnish a different proof of Hirshberg’s recent result that C–correspondences (with faithful left action) admit a (faithful) essential (that is, nondegenerate) representation on a Hilbert space, and we add that for W–correspondences this representation can be chosen normal. This work is supported by research fonds of the Department S.E.G.e S. of University of Molise.