On Spectral Triples on Crossed Products Arising From Equicontinuous Actions

Abstract

The external Kasparov product is used to construct odd and even spectral triples on crossed products of C * -algebras by actions of discrete groups which are equicontinuous in a natural sense. When the group in question is Z this gives another viewpoint on the spectral triples introduced by Belissard, Marcolli and Reihani. We investigate the properties of this construction and apply it to produce spectral triples on the Bunce-Deddens algebra arising from the odometer action on the Cantor set and some other crossed products of AF-algebras.2000 Mathematics Subject Classification. Primary: 46L05; Secondary: 46L87, 58B34.

Codes as Fractals and Noncommutative Spaces

“…Some of the actions of G C on Λ C described here are especially suitable for the crossed product construction, as shown for instance in the recent paper [9]. This means that one can regard the crossed product C(Λ C ) G C as a a spectral triple (a noncommutative manifold) and apply to it methods of noncommutative Riemannian geometry.…”

Spectral triples for noncommutative solenoidal spaces from self-coverings

“…The compact resolvent property follows. The formula for the dimension can be obtained as in [22,Thm.2.7].…”

“…. , g p ) ∈ G n (ℓ n is indeed a proper translation bounded matrix-valued function, [22,Remark 2.15]). Let us observe that G n ⊂ G n+1 and that |G n /G n−1 | = |det B| =: r. Let us define the action ρ (n) of G n on Z as follows:…”

Spectral Metric Spaces on Extensions of C*-Algebras

“…constructing new spectral triples from old ones, which is also in the sprit of permanence properties. This point of view has been used by various authors to attempt to construct spectral triples on crossed products of C*-algebras by certain discrete groups ( [2], [22]). More specifically, the authors of those two references study C * -dynamical systems (A, G, α) in which the algebra A is equipped with the structure of a spectral triple with good metric properties and consider under what conditions it is possible to write down a spectral triple on A⋊ r,α G using a natural implementation of the external product in Kasparov theory.…”