Journal article
Free probability induced by electric resistance networks on energy Hilbert spaces
Rocznik Akademii Górniczo-Hutniczej im. Stanisława Staszica. Opuscula Mathematica, Vol.31(4), pp.549-598
2011
DOI: 10.7494/OpMath.2011.31.4.549
Appears in Diamond Open Access
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.
Details
- Title: Subtitle
- Free probability induced by electric resistance networks on energy Hilbert spaces
- Creators
- Ilwoo ChoPalle Jorgensen
- Resource Type
- Journal article
- Publication Details
- Rocznik Akademii Górniczo-Hutniczej im. Stanisława Staszica. Opuscula Mathematica, Vol.31(4), pp.549-598
- DOI
- 10.7494/OpMath.2011.31.4.549
- ISSN
- 1232-9274
- Language
- English
- Date published
- 2011
- Academic Unit
- Mathematics
- Record Identifier
- 9983985982902771
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