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

TL;DR

This paper demonstrates that the stable C*-algebra of certain Smale spaces, including Williams solenoids, can be expressed as a stationary inductive limit of Fell algebras, facilitating K-theory computations.

Contribution

It establishes that the stable C*-algebra of irreducible Smale spaces with totally disconnected stable sets is a stationary inductive limit of Fell algebras, providing a new structural understanding.

Findings

The stable C*-algebra is an inductive limit of Fell algebras with compact spectrum.

This structure applies to Williams solenoids and similar examples.

The approach enables potential computation of the K-theory for these algebras.

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*-algebra is the stationary inductive limit of a C*-stable Fell algebra that has 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*-algebra. A specific one-dimensional Smale space (the aab/ab-solenoid) is considered as an illustrative running example throughout.