@Article{Cho2011,
author = "Cho, Ilwoo and Jorgensen, Palle E. T.",
title = "Free probability induced by electric resistance networks on energy Hilbert spaces",
journal = "Opuscula Math.",
year = "2011",
volume = "31",
number = "4",
pages = "549--598",
doi = "http://dx.doi.org/10.7494/OpMath.2011.31.4.549",
url = "http://dx.doi.org/10.7494/OpMath.2011.31.4.549",
abstract = "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 \(H_{\mathcal{E}}\). From \(H_{\mathcal{E}}\), 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 \(H_{\mathcal{E}}\). With the use of our ERN-groupoid, we show that \(H_{\mathcal{E}}\) may be derived as a representation Hilbert space of a universal representation of a groupoid algebra \(\mathfrak{A}_G\), and we display other representations. Among our applications, we identify a free structure of \(\mathfrak{A}_G\) in terms of the energy form.",
keywords = "directed graphs, graph groupoids, electric resistance networks, ERN-groupoids, energy Hilbert spaces, ERN-algebras, free moments, free cumulants",
issn = "1232-9274"
}