A noncommutative Sierpinski Gasket

Abstract

A quantized version of the Sierpinski gasket is proposed, on purely topological grounds, as a $C^{\ast }$-algebra ${\mathcal{A}}_{\infty }$ with a suitable form of self-similarity. Several properties of ${\mathcal{A}}_{\infty }$ are studied, in particular its nuclearity, the structure of ideals as well as the description of irreducible representations and extremal traces. A harmonic structure is introduced, giving rise to a self-similar Dirichlet form $\mathcal{E}$. A spectral triple is also constructed, extending one already known for the classical gasket, from which $\mathcal{E}$ can be reconstructed. Moreover we show that ${\mathcal{A}}_{\infty }$ is a compact quantum metric space.

28 pages, accepted for publication on the Journal of Functional Analysis

References in corpus (2)

• Dynamical Systems on Spectral Metric Spaces
• Hodge-de Rham Theory on Fractal Graphs and Fractals

Cited by in corpus (1)

• Every Continuum has a Compact Universal Cover