Free Random Variables

Voiculescu, Dan; Dykema, Ken; Nica, Alexandru

doi:10.1090/crmm/001

Cited by 1,317 publications

(1,478 citation statements)

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“…Readers can check fundamental analytic-and-combinatorial free probability from [12] and [14] (and the cited papers therein). Free probability is understood as the noncommutative operator-algebraic version of classical probability theory and statistics (covering commutative cases).…”

Section: Free Probabilitymentioning

confidence: 99%

“…Equivalently, one can have a free probability space (M P , ϕ) in the sense of [12] and [14]. Definition 2.4.…”

Section: P-adic Calculus On Q Pmentioning

confidence: 99%

“…Construct now the free product C * -probability space (M P , ψ) of the system {M p,j P } p,j , (M P , ψ) def = p∈P, j∈N0 M p,j P = p∈P,j∈N0 M P , p∈P,j∈N0 ϕ p,j , (7.11) as in Section 7.1 (e.g., see [12] and [14]), where p∈P,j∈Z M P = M P M P · · · M P . .…”

Section: The Finite-adelicmentioning

confidence: 99%

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Adelic analysis and functional analysis on the finite Adele ring

Cho¹

2018

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Abstract.In this paper, we study operator theory on the * -algebra MP , consisting of all measurable functions on the finite Adele ring A Q , in extended free-probabilistic sense. Even though our * -algebra MP is commutative, our Adelic-analytic data and properties on MP are understood as certain free-probabilistic results under enlarged sense of (noncommutative) free probability theory (well-covering commutative cases). From our free-probabilistic model on A Q , we construct the suitable Hilbert-space representation, and study a C * -algebra MP generated by MP under representation. In particular, we focus on operator-theoretic properties of certain generating operators on MP .

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Section: Free Probabilitymentioning

confidence: 99%

“…Equivalently, one can have a free probability space (M P , ϕ) in the sense of [12] and [14]. Definition 2.4.…”

Section: P-adic Calculus On Q Pmentioning

confidence: 99%

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Adelic analysis and functional analysis on the finite Adele ring

Cho¹

2018

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“…In such a case, one can understand arithmetic functions as Krein-space operators via Krein-space representations (see [12] and [13]). These studies are all motivated by number-theoretic results (e.g., [3,14] and [15]), under free probability techniques (e.g., [16,17] and [19]). …”

Section: Introductionmentioning

confidence: 99%

“…And such distortions are completely characterized by groupoid actions, sometimes called the E 0 -groupoid actions induced by partial isometries on B(H). The above framed (E 0 -)groupoids Γ G induce corresponding C * -subalgebras C * (Γ G ) of B(H), investigated by dynamical system theory and free probability (e.g., [16,17] and [19]). …”

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confidence: 99%