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 let us to 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.
Highlights
Hilbert modules have many remarkable properties, and as pointed out by Lance [31] and others, apparently quite weak notions of isometry of Hilbert modules imply isomorphism of Hilbert modules
As a guide to possible applications, we consider briefly the Cuntz semigroup and the K-theory group. These sometimes rather technical objects become attractive in the setting of C*-algebraic quantum groups, where we find that there exists a product operation on Hilbert modules that is quite nice
Compact C*-algebraic quantum groups have a well-behaved Fourier transform, which can be most briefly defined by, following Van Daele [39], see [18]: bða; FðbÞÞ 1⁄4 sðabÞ; where the elements a and b belong to a C*-algebraic quantum group A, s is the left Haar weight, and bðÁ ; ÁÞ is the pairing with the dual algebra
Summary
Hilbert modules have many remarkable properties, and as pointed out by Lance [31] and others, apparently quite weak notions of isometry of Hilbert modules imply isomorphism of Hilbert modules. We apply this basic fact in several different settings. As a guide to possible applications, we consider briefly the Cuntz semigroup and the K-theory group. These sometimes rather technical objects become attractive in the setting of C*-algebraic quantum groups, where we find that there exists a product operation on Hilbert modules that is quite nice
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