Graduate Seminar

Location:  CL 305 and Live Stream

Speaker: Jeffery Xu

MSc Student supervised by Remus Floricel

Title:  Representations of Product Systems

 Zoom Link:  https://uregina-ca.zoom.us/j/92805435386?pwd=42pEMT2QL4b8W8bqIOr5Y39hln8bSI.1

Abstract:

Product systems are a type of mathematical structure introduced by Arverson to simulate the evolution of quantum systems. Mathematically, they resemble fiber bundles of Hilbert spaces over the positive real numbers \( (0,\infty)\) equipped with an abstract form of multiplication between fibers, along with additional structure. The most common strategy for working with product systems is by introducing representations. Every product system representation acts on the fiber spaces and always decomposes into essential and singular representations, special classes that differ by how the images of the fibers behave. Representations also automatically induce an \(E\)-semigroup of certain \(^*\)-endomorphisms over some algebra of bounded operators \(\mathcal{B}(\mathcal{H})\). This semigroup action on the bounded operators in turn also induces a new product system with fibers being operator Hilbert spaces. This will all be discussed here and furthermore, a novel idea known as the word representation of a product system.