Dynamical Systems on Spectral Metric Spaces

TL;DR

This paper investigates the construction of spectral triples for dynamical systems on spectral metric spaces, especially focusing on crossed product algebras and isometric automorphisms, with applications in number and coding theory.

Contribution

It provides a canonical spectral triple construction for crossed product algebras under certain automorphisms and introduces a metric bundle approach for non-isometric cases.

Findings

Spectral triples characterize metric properties of dynamical systems.

The metric bundle construction can replace non-isometric automorphisms with isometric ones.

Applications in number theory and coding theory demonstrate practical relevance.

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 AxZ, characterizing the metric properties of the dynamical system ? If $\alpha$ 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,$\alpha$) 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.