Morita Transforms of Tensor Algebras

We show that if $M$ and $N$ are $C^{\ast }$-algebras and if $E$ (resp. $F$) is a $C^{\ast }$-correspondence over $M$ (resp. $N$), then a Morita equivalence between $(E,M)$ and $(F,N)$ implements a isometric functor between the categories of Hilbert modules over the tensor algebras of ${\mathcal{T}}_{+}(E)$ and ${\mathcal{T}}_{+}(F)$. We show that this functor maps absolutely continuous Hilbert modules to absolutely continuous Hilbert modules and provides a new interpretation of Popescu’s reconstruction operator.