On a Theorem of Cooper

Abstract

The classical result of Cooper states that every pure strongly continuous semigroup of isometries $\{V_t\}_{t \geq 0}$ on a Hilbert space is unitarily equivalent to the shift semigroup on $L^{2}([0,\infty))$ with some multiplicity.
The purpose of this note is to record a proof which has an algebraic flavour. The proof is based on the groupoid approach to semigroups of isometries initiated in [8]. We also indicate how our proof can be adapted to the Hilbert module setting and gives another proof of the main result of [3].