Inclusion systems of Hilbert modules over the C^∗-Algebra of compact operators

Abstract

In this note we observe inclusion systems of Hilbert modules over the C∗-algebra of all compact operators acting on a Hilbert space. We prove that if each Hilbert C∗-module in the generated product system is strictly complete, then it is possible to construct 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 generated product system. Thereby we obtain a generalization of the result that provides the existence of a bijection between the set of all units in an inclusion system of Hilbert spaces and the set of all units in the generated product system (B. V. R. Bhat and M. Mukherjee [Inclusion systems and amalgamated products of product systems, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (2010), no. 1, 1-26]).