Pure Semigroups of Isometries on Hilbert C*-Modules

TL;DR

This paper characterizes pure strongly continuous semigroups of adjointable isometries on Hilbert C*-modules as standard right shifts, highlighting differences from classical Hilbert space results.

Contribution

It establishes a precise characterization of pure semigroups of isometries on Hilbert C*-modules and shows limitations of classical analogies through counterexamples.

Findings

Pure semigroups are standard right shifts.

Counterexamples show classical analogies do not extend.

Highlights differences between Hilbert modules and Hilbert spaces.

Abstract

We show that pure strongly continuous semigroups of adjointable isometries on a Hilbert C*-module are standard right shifts. By counter examples, we illustrate that the analogy of this result with the classical result on Hilbert spaces by Sz.-Nagy, cannot be improved further to understand arbitrary isometry semigroups of isometries in the classical way.