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

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 onedimensional Smale space (the aab/ab-solenoid) is considered as an illustrative running example throughout. 1 2 ROBIN J. DEELEY AND ALLAN YASHINSKIunstably equivalent to some point in P. On the set X u (P), we consider the stable equivalence relation ∼ s , viewed as a groupoidThe groupoid G s (P) has anétale topology, and the stable C * -algebra of (X, ϕ) is the groupoid C * -algebra C * (G s (P)).In the case where X is a Wieler solenoid, we use the inverse limit structure to define a subrelation ∼ 0 of ∼ s . There is a corresponding subgroupoid (P), and therefore G 0 (P) isétale. Building on this, we use the fact that the inverse limit is stationary to prove the following result.Theorem 0.1. There is a nested sequence ofétale subgroupoidsThis allows one to reduce the study of G s (P) to G 0 (P), which is easier to understand. To see why it is easier, first note that the space X u (P) has a natural topology, which coincides with the topology of the diagonal subspace of G s (P). Note we never consider the subspace topology of X u (P) it inherits from X, as X u (P) is dense as a subset of X. Likewise the topology defined on G s (P) is not the same as the subspace topology it inherits as a subspace of X u (P) × X u (P). However, the topology of G 0 (P) does coincide with the subspace topology from

On finitely summable Fredholm modules from Smale spaces

“…If (X, ϕ) is mixing, then S(Q), U(P ) are also simple. Also, they are C * -stable (see [23,Theorem A.2] and [41, Corollary 4.5]) and hence, for any periodic orbit…”

Self-Similar Inverse Semigroups from Wieler Solenoids

“…The stable algebras and the stable Ruelle algebras of irreducible Wieler solenoids are studied in [2][3][4].…”

“…The purpose of this work is to study groupoids of germs and tight groupoids on a certain class of Smale spaces. Wieler [1] showed that irreducible Smale spaces with totally disconnected local stable sets can be realized as stationary inverse limit systems satisfying certain conditions, now called Wieler solenoids [1][2][3][4].…”

Smale space $C^*$-algebras have nonzero projections

“…From these results in [8], Theorem 4.4 and Lemma 2.1 we deduce the next corollary. We are additionally using the fact that the C * -algebras associated with a mixing Smale space are simple (see [25,Theorem 1.3]) and stable (see [9]). Corollary 4.5.…”

“…A priori these C * -algebras depend on the choice of P , but different choices lead to Morita equivalent C * -algebras [25]. Moreover, for any choice of P , these algebras are stable in the C * -algebraic sense [9] so different choices of P lead to isomorphic algebras. As mentioned above, associated to a (mixing) Smale space there is also the homoclinic C * -algebra, which is unital and has unique trace.…”