particular class of C\*-correspondences related to weighted partial systems. In particular it is shown that unitary equivalence and bounded isomorphisms correspond to different notions of equivalences of the original data. Even more they reflect isometric and bounded, respectively, isomorphisms of the tensor algebras.

Strong Morita equivalence for C\*-correspondences
=================================================

Muhly and Solel [@MuhSol00] initiated the study of strong Morita equivalence for C\*-correspondences. Namely ${}_A E_A$ and ${}_B F_B$ are *strongly Morita equivalent* (notation, $E {\stackrel{\textup{SME}}{\thicksim}}F$) if there exists a TRO ${}_A M_B$ such that $$E \otimes_A M \simeq M \otimes_B F {\quad\text{and}\quad}M^* \otimes_A E \simeq F \otimes_B M^*.$$ Since the tensor norm is sub-multiplicative we have that strongly Morita equivalent C\*-correspondences must be non-degenerate. Furthermore strong Morita equivalence coincides with having a TRO ${}_A M_B$ such that $$E \simeq M \otimes_B F \otimes_B M^* {\quad\text{and}\quad}F \simeq  M^* \otimes_A E \otimes_A M.$$ Again $E$ and $F$ must be non-degenerate. On the other hand if $E$ and $F$ are non-degenerate then it is easy to check that the $E$-equations give the $F$-equations.

In analogy to ${\stackrel{\textup{SME}}{\thicksim}}$ on C\*-algebras, we obtain an equivalence between representations of strongly Morita equivalent C\*-correspondences.

\[P: same repn\] Let ${}_A E_A$ and ${}_B F_B$ be strongly Morita equivalent 