Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 064, 46 pages      arXiv:1401.4622
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

Non-Commutative Resistance Networks

Marc A. Rieffel
Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA

Received January 22, 2014, in final form June 10, 2014; Published online June 14, 2014

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.

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

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