The index of a subspatial product system over a Hilbert $C^*$-module -- an example

TL;DR

This paper computes the index and characterizes all units of a unique non-Fock subsystem of a product system of Hilbert C*-modules, expanding the understanding of such structures.

Contribution

It provides the first explicit calculation of the index for a non-Fock subsystem of a product system of Hilbert C*-modules.

Findings

Computed the index of a specific non-Fock subsystem

Determined all units of the subsystem

Extended the theory of product system indices

Abstract

Recently, Bhat, Liebscher and Skeide exhibited an unusual example of product system of Hilbert $C^*$ modules. Namely, a subsystem of a Fock system, that is not Fock. Later, the authors defined the index of any product system of Hilbert $C^*$-modules, generalizing earlier partial definitions. In this note we compute the index of the mentioned interesting subsystem, and also we determine all its units.