Growth rates of dimensional invariants of compact quantum groups and a theorem of Høegh-Krohn, Landstad and Størmer
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Abstract:
We give local upper and lower bounds for the eigenvalues of the modular operator associated to an ergodic action of a compact quantum group on a unital $C^*$–algebra. They involve the modular theory of the quantum group and the growth rate of quantum dimensions of its representations and they become sharp if other integral invariants grow subexponentially. For compact groups, this reduces to the finiteness theorem of Høegh-Krohn, Landstad and Størmer. Consequently, compact quantum groups of Kac type admitting an ergodic action with a non-tracial invariant state must have representations whose dimensions grow exponentially. In particular, $S_{-1}U(d)$ acts ergodically only on tracial $C^*$–algebras. For quantum groups with non-involutive coinverse, we derive a lower bound for the parameters $0<\lambda <1$ of factors of type III${}_\lambda$ that can possibly arise from the GNS representation of the invariant state of an ergodic action with a factorial centralizer.References
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Additional Information
- Claudia Pinzari
- Affiliation: Dipartimento di Matematica, Sapienza Università di Roma, 00185–Roma, Italy
- Email: pinzari@mat.uniroma1.it
- Received by editor(s): February 15, 2011
- Received by editor(s) in revised form: July 14, 2011
- Published electronically: July 5, 2012
- Communicated by: Marius Junge
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 895-907
- MSC (2010): Primary 46L55, 46L65; Secondary 37A55, 28D20
- DOI: https://doi.org/10.1090/S0002-9939-2012-11482-0
- MathSciNet review: 3003682
Dedicated: Dedicated to the memory of Claudio D’Antoni