A noncommutative Sierpinski Gasket

TL;DR

This paper introduces a quantum analog of the Sierpinski gasket as a $C^*$-algebra, exploring its properties, harmonic structure, spectral triple, and quantum metric space characteristics.

Contribution

It constructs a noncommutative Sierpinski gasket with self-similarity, analyzing its algebraic, spectral, and metric properties, extending classical fractal analysis to the quantum setting.

Findings

The algebra $ _ abla$ is nuclear and has a well-understood ideal structure.

A self-similar Dirichlet form $ _ abla$ is established.

The quantum gasket is shown to be a compact quantum metric space.

Abstract

A quantized version of the Sierpinski gasket is proposed, on purely topological grounds, as a $C^*$-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.