Pure semigroups of isometries on Hilbert C⁎C⁎-modules

We show that pure strongly continuous semigroups of adjointable isometries on a Hilbert C⁎C⁎-module are standard right shifts. By counterexamples, we illustrate that the analogy of this result with the classical result on Hilbert spaces by Cooper, cannot be improved further to understand arbitrary semigroups of isometries in the classical way. The counterexamples include a strongly continuous semigroup of non-adjointable isometries, an extension of the standard right shift that is not strongly continuous, and a strongly continuous semigroup of adjointable isometries that does not admit a decomposition into a maximal unitary part and a pure part.