On the K-theory of the stable C⁎-algebras from substitution tilings

Gonçalves, Daniel

doi:10.1016/j.jfa.2010.10.020

Cited by 7 publications

(18 citation statements)

References 13 publications

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“…We would be remiss not to mention that, for higher dimensional examples, the computation of the K -theory from the inductive limit becomes increasingly difficult. The reader can see such examples (even in dimension two) in [8,9]. Nevertheless, in future work, the present authors will use the techniques from the present paper to compute the K -theory in a number of examples.…”

Section: Theorem 12 (See Theorem 417)mentioning

confidence: 99%

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

Deeley¹,

Yashinski²

2019

Ergod. Th. Dynam. Sys.

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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

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Section: Theorem 12 (See Theorem 417)mentioning

confidence: 99%

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

Deeley¹,

Yashinski²

2019

Ergod. Th. Dynam. Sys.

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“…Finally, given anétale equivalence relation R over a locally compact set X, the groupoid algebra associated to it is obtained as the completion, over a certain norm (see [18] or [7]), of the *-algebra of the continuous functions with compact support in R, C c (R), where the *-algebra operations are defined, for f, g ∈ C c (R),…”

Section: Introductionmentioning

confidence: 99%

Partial crossed products as equivalence relation algebras

Beuter¹

2016

Rocky Mountain J. Math.

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For a free partial action of a group in a set we realize the associated partial skew group ring as an algebra of functions with finite support over an equivalence relation and we use this result to characterize the ideals in the partial skew group ring. This generalizes, to the purely algebraic setting, the known characterization of partial C*-crossed products as groupoid C*-algebras. For completeness we include a new proof of the C* result for free partial actions.

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“…In the case that L is repetitive, aperiodic and has finite local complexity, one can characterise Ω L as a projective limit [2,50,12] and compute its K-theory using the Pimsner-Voiculescu spectral sequence [87] (adapted from the spectral sequence used by Kasparov [45, §6.10]), whose E 2 -page is isomorphic to theČech cohomology of Ω L with integer coefficients. In the case of low-dimensional substitution tilings with finite local complexity and a primitive and injective substitution map, Gonçalves-Ramirez-Solano relate theČech cohomology of Ω L to the K-theory of the groupoid C * -algebra of the unstable equivalence relation on Ω L (note that this groupoid C * -algebra is Morita equivalent to C * r (G)) [39,Theorem 2.3]. See [39] for a detailed exposition on these (and other) matters.…”

Section: Delone Sets and The Transversal Groupoidmentioning

confidence: 99%

“…In the case of low-dimensional substitution tilings with finite local complexity and a primitive and injective substitution map, Gonçalves-Ramirez-Solano relate theČech cohomology of Ω L to the K-theory of the groupoid C * -algebra of the unstable equivalence relation on Ω L (note that this groupoid C * -algebra is Morita equivalent to C * r (G)) [39,Theorem 2.3]. See [39] for a detailed exposition on these (and other) matters.…”

Section: Delone Sets and The Transversal Groupoidmentioning

confidence: 99%

Index Theory and Topological Phases of Aperiodic Lattices

Bourne

Mesland

2019

Ann. Henri Poincaré

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We examine the noncommutative index theory associated to the dynamics of a Delone set and the corresponding transversal groupoid. Our main motivation comes from the application to topological phases of aperiodic lattices and materials, and applies to invariants from tilings as well. Our discussion concerns semifinite index pairings, factorisation properties of Kasparov modules and the construction of unbounded Fredholm modules for lattices with finite local complexity.

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On the K-theory of the stable C⁎-algebras from substitution tilings

Cited by 7 publications

References 13 publications

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

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

Partial crossed products as equivalence relation algebras

Index Theory and Topological Phases of Aperiodic Lattices

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