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

Published online by Cambridge University Press:  10 April 2019

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

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.

Type
Original Article
Copyright
© Cambridge University Press, 2019

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