Induced representations of Hilbert modules over locally \(C^*\)-algebras and the imprimitivity theorem (Q2815912)

\textit{M. Joiţa} and the reviewer [Stud. Math. 209, No. 1, 11--19 (2012; Zbl 1272.46047)] introduced a notion of Morita equivalence in the category of Hilbert \(C^*\)-modules. The authors of the present paper use it to study induced representations of Hilbert modules over locally \(C^*\)-algebras and their non-degeneracy. They prove that two full Hilbert modules over locally \(C^*\)-algebras are Morita equivalent if and only if their underlying locally \(C^*\)-algebras are strongly Morita equivalent and then they give a module version of the imprimitivity theorem. They indeed prove that, for Morita equivalent Hilbert modules \(V\) and \(W\) over locally \(C^*\)-algebras \(A\) and \(B\), respectively, there is a bijective correspondence between equivalence classes of non-degenerate representations of \(V\) and \(W\).