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The stable algebra of a Wieler solenoid: inductive limits and $K$-theory

Part of: Selfadjoint operator algebras Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  10 April 2019

ROBIN J. DEELEY  and
ALLAN YASHINSKI
ROBIN J. DEELEY
Affiliation:
Department of Mathematics, University of Colorado Boulder, Campus Box 395, Boulder, CO80309-0395, USA email robin.deeley@colorado.edu
ALLAN YASHINSKI
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD20742-4015, USA email ayashins@math.umd.edu
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Abstract

Wieler has shown that every irreducible Smale space with totally disconnected stable sets is a solenoid (i.e., obtained via a stationary inverse limit construction). Using her construction, we show that the associated stable $C^{\ast }$-algebra is the stationary inductive limit of a $C^{\ast }$-stable Fell algebra that has a compact spectrum and trivial Dixmier–Douady invariant. This result applies in particular to Williams solenoids along with other examples. Beyond the structural implications of this inductive limit, one can use this result to, in principle, compute the $K$-theory of the stable $C^{\ast }$-algebra. A specific one-dimensional Smale space (the $aab/ab$-solenoid) is considered as an illustrative running example throughout.

Keywords

operator algebrastopological dynamicsétale groupoidssolenoids

MSC classification

Primary: 46L80: $K$-theory and operator algebras (including cyclic theory) 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 10 , October 2020 , pp. 2734 - 2768
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Copyright
© Cambridge University Press, 2019

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