Inclusion systems of Hilbert modules over the \(C^\ast\)-algebra of compact operators (Q2831915)

The paper introduces the concept of inclusion systems of Hilbert \(C^*\)-modules generalizing the notion of inclusion systems of Hilbert spaces. More precisely, the paper is concerned with inclusion systems of Hilbert \(\mathcal{B}\)-\(\mathcal{B}\) bimodules, where \(\mathcal{B}\) denotes the \(C^*\)-algebra of compact operators acting on a Hilbert space \(H\). The very special nature of \(\mathcal{B}\) as the underlying algebra of coefficients provides, via systematic use of the supporting submodule over the Hilbert-Schmidt class, a bridge to certain Hilbert space results which are used in the proof of the main result.NEWLINENEWLINEThe first result of the paper states that each inclusion system of Hilbert \(\mathcal{B}\)-\(\mathcal{B}\) bimodules generates, by taking a suitable inductive limit, an associated product system of Hilbert \(\mathcal{B}\)-\(\mathcal{B}\) bimodules. The main result of the paper, Theorem 3, states that, provided that each Hilbert module in the associated product system is strictly complete, there is a bijection between the set of all units of an inclusion system and a quotient (by a suitable equivalence relation) of a certain set of units in the associated product system. This serves as a generalization of a result proved by B. V. R. Bhat and M. Mukherjee for inclusion systems of Hilbert spaces.