Opuscula Math. 31, no. 4 (2011), 549-598
http://dx.doi.org/10.7494/OpMath.2011.31.4.549

Opuscula Mathematica

# Free probability induced by electric resistance networks on energy Hilbert spaces

Ilwoo Cho
Palle E. T. Jorgensen

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.

Mathematics Subject Classification: 05C62, 05C90, 17A50, 18B40, 47A99.

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• Ilwoo Cho
• St. Ambrose University, Department of Mathematics, 518 W. Locust St., Davenport, Iowa, 52803, USA
• Palle E. T. Jorgensen
• The University of Iowa, Department of Mathematics, 14 McLean Hall, Iowa City, Iowa, 52242, USA
• Revised: 2011-03-29.
• Accepted: 2011-03-29.