Roots in product systems

Abstract

We observe the time ordered product system IΓ⊗(F), 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. We discuss the notion of roots therein and we present the form of all continuous roots of the vacuum unit ω. Also, we look at the relation between F and the set Rω (the set of all continuous roots of ω in IΓ⊗(F)). We state that the continuous roots of ω in the time ordered product system IΓ⊗(F) are indexed by the elements of F itself, via the isomorphism between Hilbert left-right modules F and Rω.