Additive Units of Inclusion Systems of Hilbert Modules over the $$C^*$$-Algebra of Compact Operators

Abstract

We introduce the notion of additive units (addits) of a pointed inclusion system of Hilbert modules over the \(C^*\)-algebra of all compact operators acting on a Hilbert space G. By a pointed inclusion system, we mean an inclusion system with a fixed normalised reference unit. We prove that if G is a Hilbert space of finite dimension, then there is a bijection between the set of addits of a pointed inclusion system and the set of addits of the generated product system. We also consider addits of spatial product systems of Hilbert modules and, as an example, we find all continuous addits in the product system from Barreto et al. (J Funct Anal 212:121–181, 2004, Example 4.2.4).