Morita equivalence in the context of Hilbert modules (Q2782224)

A quantale is an involutive complete lattice \(A\) with an associative multiplication \(\cdot\) satisfyingNEWLINE\(x\cdot\bigvee_{\iota\in I}x_\iota= \bigvee_{\iota\in I}x\cdot x_\iota\) for \(x,x_\iota\in A\). The classical Morita equivalence was established for quantales in [\textit{F. Borceux} and \textit{E. M. Vitale} [Suppl. Rend. Circ. Mat. Palermo, II. Ser. 29, 353--362 (1992; Zbl 0781.06013)], by replacing the category of Abelian groups with the symmetric monoidal closed category of complete semilattices. The notion of a monoid is meaningful therein and so the structure is rich enough to consider categories of bimodules. A Morita theory was also developed for Hilbert \(C^*\)-modules over (non-unital) \(C^*\)-algebras. Motivated by and combining these, the author extends the main result to non-unital quantales.NEWLINENEWLINEFor the entire collection see [Zbl 0983.00046].