We show that if M and N are C*-algebras and if E (resp.F) is a C*-correspondence over M (resp. N), then aMorita equivalence between (E,M) and (F,N) implements an isometricfunctor between the categories of Hilbert modules over the tensoralgebras of T+(E) and T+(F). We showthat this functor maps absolutely continuous Hilbert modules to absolutelycontinuous Hilbert modules and provides a new interpretation of Popescu'sreconstruction operator.
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