Free probability induced by electric resistance networks on energy Hilbert spaces

• Autorzy: Cho I. , Jorgensen P. E. T.
• Języki publikacji: EN

Abstrakty

EN

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 Hε. From Hε, 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 Hε. With the use of our ERN-groupoid, we show that Hε may be derived as a representation Hilbert space of a universal representation of a groupoid algebra [formula], and we display other representations. Among our applications, we identify a free structure of [formula] in terms of the energy form.

Słowa kluczowe

EN

directed graphs; graph groupoids; electric resistance networks; ERN-groupoids; energy Hilbert spaces; ERN-algebras; free moments; free cumulants

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2011
• Tom: Vol. 31, no. 4
• Strony: 549--598
• Opis fizyczny: Bibliogr. 33 poz.

Twórcy

autor

Cho I.

autor

Jorgensen P. E. T.

• St. Ambrose University Department of Mathematics 518 W. Locust St., Davenport, Iowa, 52803, USA,

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