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1

Banach *-algebras generated by semicircular elements induced by certain orthogonal projections

Cho I., Jorgensen P. E. T.

Opuscula Mathematica

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2018

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Vol. 38, no. 4

501--535

EN

The main purpose of this paper is to study structure theorems of Banach *-algebras generated by semicircular elements. In particular, we are interested in the cases where given semicircular elements are induced by orthogonal projections in a C*-probability space.

2

Semicircular elements induced by p-adic number fields

Cho I., Jorgensen P. E. T.

Opuscula Mathematica

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2017

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Vol. 37, no. 5

665--703

EN

In this paper, we study semicircular-like elements, and semicircular elements induced by p-adic analysis, for each prime p. Starting from a p-adic number field Qp, we construct a Banach *-algebra [formula], for a fixed prime p, and show the generating elements Qpj of [formula] form weighted-semicircular elements, and the corresponding scalar-multiples Θpj of Qpj become semicircular elements, for all j ∈ Z. The main result of this paper is the very construction of suitable linear functionals [formula] on [formula], making Qpj be weighted-semicircular, for all j ∈ Z.

3

Free probability induced by electric resistance networks on energy Hilbert spaces

Cho I., Jorgensen P. E. T.

Opuscula Mathematica

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2011

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Vol. 31, no. 4

549-598

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.

4

A sampling theory for infinite weighted graphs

Jorgensen P. E. T.

Opuscula Mathematica

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2011

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Vol. 31, no. 2

209-236

EN

We prove two sampling theorems for infinite (countable discrete) weighted graphs G; one example being "large grids of resistors" i.e., networks and systems of resistors. We show that there is natural ambient continuum X containing G, and there are Hilbert spaces of functions on X that allow interpolation by sampling values of the functions restricted only on the vertices in G. We sample functions on X from their discrete values picked in the vertex-subset G. We prove two theorems that allow for such realistic ambient spaces X for a fixed graph G, and for interpolation kernels in function Hilbert spaces on X, sampling only from points in the subset of vertices in G. A continuum is often not apparent at the outset from the given graph G. We will solve this problem with the use of ideas from stochastic integration.