Gap-labelling for three-dimensional aperiodic solids

doi:10.1016/S0764-4442(01)01892-4

Comptes Rendus de l'Académie des Sciences - Series I - Mathematics

Volume 332, Issue 6, 15 March 2001, Pages 521-525

Comptes Rendus de l'...

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Abstract

Let X be totally disconnected compact Hausdorff space with an action α of $Z^{3}$ by commuting homeomorphisms and with an invariant probability measure μ. We show that $τ_{∗} K_{0} (C(X)⋊_{α} Z^{3})=μ(C(X, Z))$ where τ is the trace induced on the crossed product $C(X)⋊_{α} Z^{3}$ .

Résumé

Soit X un espace topologique compact séparé, totalement discontinu, muni d'une action α de $Z^{3}$ par trois homéomorphismes commutant mutuellement, pour lesquels μ est une mesure de probabilité invariante. Il est alors démontré que $τ_{∗} K_{0} (C(X)⋊_{α} Z^{3})=μ C(X, Z)$ , expression dans laquelle τ est la trace sur le produit croisé $C(X)⋊_{α} Z^{3}$ induite par μ.

References (11)

J. Bellissard
Gap labelling theorems for Schrödinger's operators
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Hulls of aperiodic solids and gap-labelling theorems
B. Blackadar
K-Theory for Operator Algebras
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A. Connes
Cyclic cohomology and the transverse fundamental class of a foliation
A. Connes
Non Commutative Geometry
(1994)

There are more references available in the full text version of this article.

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Gap-labelling for three-dimensional aperiodic solidsLe théorème de l'étiquetage des gaps pour les solides apériodiques de dimension 3

Abstract

Résumé

Gap labelling theorems for Schrödinger's operators

Hulls of aperiodic solids and gap-labelling theorems

K-Theory for Operator Algebras

Cyclic cohomology and the transverse fundamental class of a foliation

Non Commutative Geometry

Gap-labelling for three-dimensional aperiodic solidsLe théorème de l'étiquetage des gaps pour les solides apériodiques de dimension $3$