Non-Commutative Resistance Networks

Marc A. Rieffel

Symmetry, integrability and geometry: methods and applications, Tome 10 (2014)

Résumé

In the setting of finite-dimensional $C^*$-algebras ${\mathcal A}$ we define what we call a Riemannian metric for ${\mathcal A}$, which when ${\mathcal A}$ is commutative is very closely related to a finite resistance network. We explore the relationship with Dirichlet forms and corresponding seminorms that are Markov and Leibniz, with corresponding matricial structure and metric on the state space. We also examine associated Laplace and Dirac operators, quotient energy seminorms, resistance distance, and the relationship with standard deviation.

Détail

Keywords: resistance network; Riemannian metric; Dirichlet form; Markov; Leibniz seminorm; Laplace operator; resistance distance; standard deviation.

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