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\iteman{io-port 05957037}
\itemau{Cho, Ilwoo; Jorgensen, Palle E.T.}
\itemti{Free probability induced by electric resistance networks on energy Hilbert spaces.}
\itemso{Opusc. Math. 31, No. 4, 549-598 (2011).}
\itemab
Summary: 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_\epsilon $. From $H_\epsilon $, 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_\epsilon $. With the use of our ERN-groupoid, we show that $H_\epsilon $ may be derived as a representation Hilbert space of a universal representation of a groupoid algebra $\frak A_G$, and we display other representations. Among our applications, we identify a free structure of $\frak A_G$ in terms of the energy form.
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\itemut{directed graphs; graph groupoids; electric resistance networks; ERN-groupoids; energy Hilbert spaces; ERN-algebras; free moments; free cumulants}
\itemli{doi:10.7494/OpMath.2011.31.4.549}
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