Co-actions, Isometries, and isomorphism classes of Hilbert modules
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Tusi Mathematical Research Group
ORIGINAL PAPER
Co-actions, Isometries, and isomorphism classes of Hilbert modules Dan Z. Kucˇerovsky´1 Received: 14 July 2020 / Accepted: 21 July 2020 Ó The Author(s) 2020
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. Keywords Hilbert modules Cuntz semigroups C*-algebraic quantum group Multiplicative unitaries
Mathematics Subject Classification 47L80 16T05 47L50 16T20
1 Introduction 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 particularly 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 Communicated by Evgenij Troitsky. & Dan Z. Kucˇerovsky´ [email protected] 1
University of New Brunswick at Fredericton, Fredericton E3B 5A3, Canada
D. Z. Kucˇerovsky´
and seems to be distinct from, although related to, the usual interior and exterior tensor products of Hilbert modules. We then study isomorphism classes of Hilbert modules. These have a natural semigroup structure, under direct sum. Our most advanced result is roughly as follows: Theorem 1 Let the C* -algebraic quantum group A have a left co-action upon B with a faithful invariant state. Then, we obtain a well-behaved action of the semiring of isomorphism classes of Hilbert modules over A upon the semigroup of isomorphism classes of Hilbert modules over B. In the above, by an invariant state for a left co-action d of a compact or discrete C*-algebraic quantum group ðA; DÞ on B we mean a state g on B such that ðId gÞd ¼ gðÞ1: By a well-behaved action of the semiring, we mean a multiplicative action where the product on the ring distributes over the direct sum in the semigroup. As well as providing some kind of invariant associated with a co-action, it is quite possible that the above can be viewed as an algebraic topology type of obstacle to the existence of certain co-actions. In other words, we could show nonexistence of certain co-actions by showing that one of the implied equations for the action of the semiring on the semigroup must fail. The work of Goswami [17] on non-existence of certain co-actions pro
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