On the set of addits in time ordered product systems

Abstract

We observe the time ordered product system IΓ⊗(F) = (IΓt(F))t∈R+ , where F is a two-sided Hilbert module over the C∗-algebra B of all bounded operators acting on a Hilbert space of finite dimension. It has a central unital unit - the vacuum unit ω = (ωt)t∈R+ , so it is a spatial product system. Therein we consider additive units (addits) as additive counterparts to the multiplicative notion of units. We consider continuous addits in particular and we discuss their properties. The set of all continuous addits of ω in IΓ⊗(F) is denoted by Aω. We show that Aω and F ⊕ B are isomorphic as Hilbert B − B modules.