Opuscula Math. 31, no. 4 (2011), 549-598

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
  • Received: 2011-01-06.
  • Revised: 2011-03-29.
  • Accepted: 2011-03-29.
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Cite this article as:
Ilwoo Cho, Palle E. T. Jorgensen, Free probability induced by electric resistance networks on energy Hilbert spaces, Opuscula Math. 31, no. 4 (2011), 549-598, http://dx.doi.org/10.7494/OpMath.2011.31.4.549

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