%0 Journal Article
%T Free probability induced by electric resistance networks on energy Hilbert spaces
%A Ilwoo Cho
%A Palle E.T. Jorgensen
%J Opuscula Mathematica
%D 2011
%I AGH University of Science and Technology
%X We show that a class of countable weighted graphs arising in the study of electric resistance networks (ERNs) are naturally associated with groupoids. Starting with a fixed ERN, it is known that there is a canonical energy form and a derived energy Hilbert space $H_E$. From $H_E$, one then studies resistance metrics and boundaries of the ERNs. But in earlier research, there does not appear to be a natural algebra of bounded operators acting on $H_E$. With the use of our ERN-groupoid, we show that $H_E$ may be derived as a representation Hilbert space of a universal representation of a groupoid algebra $A_G$, and we display other representations. Among our applications, we identify a free structure of $A_G $ in terms of the energy form.
%K directed graphs
%K graph groupoids
%K electric resistance networks
%K ERN-groupoids
%K energy Hilbert spaces
%K ERN-algebras
%K free moments
%K free cumulants
%U http://www.opuscula.agh.edu.pl/vol31/4/art/opuscula_math_3138.pdf