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Dynamical Systems on Spectral Metric Spaces

Bellissard, Jean V. and Marcolli, Matilde and Reihani, Kamran (2010) Dynamical Systems on Spectral Metric Spaces. . (Submitted)

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Let (A,H,D) be a spectral triple, namely: A is a C^*-algebra, H is a Hilbert space on which A acts and D is a selfadjoint operator with compact resolvent such that the set of elements of A having a bounded commutator with D is dense. A spectral metric space, the noncommutative analog of a complete metric space, is a spectral triple (A,H,D) with additional properties which guaranty that the Connes metric induces the weak^*-topology on the state space of A. A ^*-automorphism respecting the metric defined a dynamical system. This article gives various answers to the question: is there a canonical spectral triple based upon the crossed product algebra Ax_αZ, characterizing the metric properties of the dynamical system ? If α is the noncommutative analog of an isometry the answer is yes. Otherwise, the metric bundle construction of Connes and Moscovici is used to replace (A,α) by an equivalent dynamical system acting isometrically. The difficulties relating to the non compactness of this new system are discussed. Applications, in number theory, in coding theory are given at the end.

Item Type:Report or Paper (Discussion Paper)
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Additional Information:(Submitted on 26 Aug 2010) Work supported in part by NSF Grants No. 0901514, DMS-0651925 and DMS-1007207.
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Record Number:CaltechAUTHORS:20170712-092526227
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:79001
Deposited By: Ruth Sustaita
Deposited On:12 Jul 2017 16:38
Last Modified:03 Oct 2019 18:14

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