Co-actions, Isometries and isomorphism classes of Hilbert modules

TL;DR

This paper characterizes when A-linear maps between Hilbert A-modules are induced by unitaries, and applies this to classify modules over quantum groups, with implications for co-actions and the Cuntz semigroup functor.

Contribution

It develops a criterion for A-linear maps to be unitarily induced and applies it to classify Hilbert modules over quantum groups, advancing the understanding of co-actions.

Findings

A-linear maps extend to unitaries on enveloping spaces if and only if induced by a Hilbert module operator.

Classifies Hilbert modules over certain C*-algebraic quantum groups via isometry classes.

Shows the Cuntz semigroup functor translates co-actions into multiplicative actions.

Abstract

We show that a A-linear map of Hilbert A-modules is induced by a unitary Hilbert module operator if and only if it extends to an ordinary unitary on appropriately defined enveloping Hilbert spaces. Applications to the theory of multiplicative unitaries compute the equivalence classes of Hilbert modules over a class of C*- algebraic quantum groups. We thus develop a theory that for example could be used to show non-existence of certain co-actions. In particular, we show that the Cuntz semigroup functor takes a co-action to a multiplicative action.