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 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.

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• The Podles spheres converge to the sphere
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• Spectral metric spaces for Gibbs measures
• Operator *-correspondences in analysis and geometry
• A $P$-Adic Spectral Triple
• Lifting spectral triples to noncommutative principal bundles
• Contractive spectral triples for crossed products
• Fractal spectral triples on Kellendonk's $C^{\ast }$-algebra of a substitution tiling
• Noncommutative geometry of the Moyal plane: translation isometries, Connes' distance on coherent states, Pythagoras equality
• A Spectral Triple for a Solenoid Based on the Sierpinski Gasket
• Real Spectral Triples on Crossed Products
• A classification of aperiodic order via spectral metrics and Jarník sets
• A noncommutative Sierpinski Gasket
• Sup-norm-closable bilinear forms and Lagrangians
• A characterization of subshifts with bounded powers
• Spectral triples on irreversible $C^{\ast }$-dynamical systems
• Codes as fractals and noncommutative spaces
• Topographic Gromov-Hausdorff quantum Hypertopology for Quantum Proper Metric Spaces