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Dynamics of compact quantum metric spaces

Part of: Selfadjoint operator algebras Linear spaces and algebras of operators

Published online by Cambridge University Press:  11 May 2020

JENS KAAD Open the ORCID record for JENS KAAD [Opens in a new window]  and
DAVID KYED Open the ORCID record for DAVID KYED [Opens in a new window]
JENS KAAD
Affiliation:
Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej55, DK-5230 Odense M, Denmark email kaad@imada.sdu.dk, dkyed@imada.sdu.dk
DAVID KYED
Affiliation:
Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej55, DK-5230 Odense M, Denmark email kaad@imada.sdu.dk, dkyed@imada.sdu.dk
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Abstract

We provide a detailed study of actions of the integers on compact quantum metric spaces, which includes general criteria ensuring that the associated crossed product algebra is again a compact quantum metric space in a natural way. Moreover, we provide a flexible set of assumptions ensuring that a continuous family of $\ast$-automorphisms of a compact quantum metric space yields a field of crossed product algebras which varies continuously in Rieffel’s quantum Gromov–Hausdorff distance. Finally, we show how our results apply to continuous families of Lip-isometric actions on compact quantum metric spaces and to families of diffeomorphisms of compact Riemannian manifolds which vary continuously in the Whitney $C^{1}$-topology.

Keywords

46L05quantum metric spacescrossed productsdynamics of Riemannian manifolds

MSC classification

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 46L89: Other “noncommutative” mathematics based on $C^*$-algebra theory 47L65: Crossed product algebras (analytic crossed products)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 7 , July 2021 , pp. 2069 - 2109
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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