Dynamical Systems on Spectral Metric Spaces

Abstract

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.

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