Non-Commutative Resistance Networks

Symmetry Integrability and Geometry-Methods and Applications

article

Marc A. Rieffel¹ 

[1] Department of Mathematics, University of California

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

DOI  :  10.3842/SIGMA.2014.064

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown