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124
qGaussian processes: Noncommutative and classical aspects
 Commun. Math. Phys
, 1997
"... Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expec ..."
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Cited by 79 (3 self)
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Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation state. We show that there is a qanalogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on qGaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of qGaussian processes possess a noncommutative kind of Markov property, which ensures that there exist classical versions of these noncommutative processes. This answers an old question of Frisch and Bourret [FB].
Convolution And Limit Theorems For Conditionally Free Random Variables
"... We introduce the notion of a conditionally free product and conditionally free convolution. We describe this convolution both from a combinatorial point of view, by showing its connection with the lattice of noncrossing partitions, and from an analytic point of view, by presenting the basic formula ..."
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Cited by 59 (1 self)
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We introduce the notion of a conditionally free product and conditionally free convolution. We describe this convolution both from a combinatorial point of view, by showing its connection with the lattice of noncrossing partitions, and from an analytic point of view, by presenting the basic formula for its Rtransform. We calculate
Free Probability Theory And NonCrossing Partitions
 LOTHAR. COMB
, 1997
"... Voiculescu's free probability theory  which was introduced in an operator algebraic context, but has since then developed into an exciting theory with a lot of links to other fields  has an interesting combinatorial facet: it can be described by the combinatorial concept of multiplicat ..."
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Cited by 48 (4 self)
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Voiculescu's free probability theory  which was introduced in an operator algebraic context, but has since then developed into an exciting theory with a lot of links to other fields  has an interesting combinatorial facet: it can be described by the combinatorial concept of multiplicative functions on the lattice of noncrossing partitions. In this survey I want to explain this connection  without assuming any knowledge neither on free probability theory nor on noncrossing partitions.
Interpolated free group factors
 Pacific J. Math
, 1994
"... Abstract. The interpolated free group factors L(Fr) for 1 < r ≤ ∞, (also defined by F. Rădulescu) are given another (but equivalent) definition as well as proofs of their properties with respect to compression by projections and free products. In order to prove the addition formula for free produ ..."
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Cited by 42 (4 self)
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Abstract. The interpolated free group factors L(Fr) for 1 < r ≤ ∞, (also defined by F. Rădulescu) are given another (but equivalent) definition as well as proofs of their properties with respect to compression by projections and free products. In order to prove the addition formula for free products, algebraic techniques are developed which allow us to show R∗R ∼ = L(F2) where R is the hyperfinite II1–factor. Introduction. The free group factors L(Fn) for n = 2, 3,..., ∞ (introduced in [4]) have recently been extensively studied [11,2,5,6,7] using Voiculescu’s theory of freeness in noncommutative probability spaces (see [8,9,10,11,12,13], especially the latter for an overview). One hopes to eventually be able to solve the old isomorphism question, first raised by R.V. Kadison in the 1960’s, of whether L(Fn) ∼ = L(Fm) for n ̸ = m. In [7], F. Rădulescu introduced
Free Quasifree States
, 1997
"... To a real Hilbert space and a oneparameter group of orthogonal transformations we associate a C∗algebra which admits a free quasifree state. This construction is a freeprobability analog of the construction of quasifree states on the CAR and CCR algebras. We show that under certain conditions, o ..."
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Cited by 40 (8 self)
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To a real Hilbert space and a oneparameter group of orthogonal transformations we associate a C∗algebra which admits a free quasifree state. This construction is a freeprobability analog of the construction of quasifree states on the CAR and CCR algebras. We show that under certain conditions, our C∗algebras are simple, and the free quasifree states are unique. The corresponding von Neumann algebras obtained via the GNS construction are free analogs of the ArakiWoods factors. Such von Neumann algebras can be decomposed into free products of other von Neumann algebras. For nontrivial oneparameter groups, these von Neumann algebras are type III factors. In the case the oneparameter group is nontrivial and almostperiodic, we show that Connes’ Sd invariant completely classifies these algebras.
Stable Laws and Domains of Attraction in Free Probability Theory
, 1999
"... In this paper we determine the distributional behavior of sums of free (in the sense of Voiculescu) identically distributed, infinitesimal random variables. The theory is shown to parallel the classical theory of independent random variables, though the limit laws are usually quite di#erent. Our wor ..."
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Cited by 39 (0 self)
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In this paper we determine the distributional behavior of sums of free (in the sense of Voiculescu) identically distributed, infinitesimal random variables. The theory is shown to parallel the classical theory of independent random variables, though the limit laws are usually quite di#erent. Our work subsumes all previously known instances of weak convergence of sums of free, identically distributed random variables. In particular, we determine the domains of attraction of stable distributions in the free theory. These freely stable distributions are studied in detail in the appendix, where their unimodality and duality properties are demonstrated.
Brown's Spectral Distribution Measure for Rdiagonal Elements in Finite von Neumann Algebras
 Neumann Algebras, Journ. Functional Analysis
, 1999
"... In 1983 L. G. Brown introduced a spectral distribution measure for nonnormal elements in a finite von Neumann algebra M with respect to a fixed normal faithful tracial state . In this paper we compute Brown's spectral distribution measure in case T has a polar decomposition T = UH where U is ..."
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Cited by 39 (8 self)
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In 1983 L. G. Brown introduced a spectral distribution measure for nonnormal elements in a finite von Neumann algebra M with respect to a fixed normal faithful tracial state . In this paper we compute Brown's spectral distribution measure in case T has a polar decomposition T = UH where U is a Haar unitary and U and H are free. (When Ker T = f0g this is equivalent to that (T ; T ) is an Rdiagonal pair in the sense of Nica and Speicher.) The measure T is expressed explicitly in terms of the Stransform of the distribution T T of the positive operator T T . In case T is a circular element, i.e., T = (X 1 + iX 2 )= p 2 where (X 1 ; X 2 ) is a free semicircular system, then sp T = D, the closed unit disk, and T has constant density 1= on D. 1 Introduction In 1995 Nica and Speicher introduced the class of Rdiagonal pairs in noncommutative probability spaces (see [10]). A pair (a; b) in the noncommutative probability space (A; ') is called Rdiagonal if the (2dime...
Free products of hyperfinite von Neumann algebras and free dimension
 DUKE MATH. J
, 1992
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Embeddings of reduced free products of operator algebras
 Pacific J. Math. 199
, 2001
"... Abstract. Given reduced amalgamated free products of C ∗ –algebras (A, φ) = ∗ ι∈I (Aι, φι) and (D, ψ) = ∗ ι∈I (Dι, ψι), an embedding A ↩ → D is shown to exist assuming there are conditional expectation preserving embeddings Aι ↩ → Dι. This result is extended to show the existence of the reduced am ..."
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Cited by 30 (10 self)
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Abstract. Given reduced amalgamated free products of C ∗ –algebras (A, φ) = ∗ ι∈I (Aι, φι) and (D, ψ) = ∗ ι∈I (Dι, ψι), an embedding A ↩ → D is shown to exist assuming there are conditional expectation preserving embeddings Aι ↩ → Dι. This result is extended to show the existence of the reduced amalgamated free product of certain classes of unital completely positive maps. Finally, the reduced amalgamated free product of von Neumann algebras is defined in the general case and analogues of the above mentioned results are proved for von Neumann algebras. Introduction. We begin with some standard facts about freeness in groups, analogues of which we will consider in C ∗ –algebras. If H is a subgroup of a group G and if Gι is a subgroup of G containing H for every ι in some index set I, let us say that the family (Gι)ι∈I is free over H (or free with amalgamation over H) if g1g2 · · · gn / ∈ H whenever gj ∈ Gιj \H for some ιj ∈ I
DT–operators and decomposability of Voiculescu’s circular operator
, 2002
"... The DT–operators are introduced, one for every pair (µ, c) consisting of a compactly supported Borel probability measure µ on the complex plane and a constant c> 0. These are operators on Hilbert space that are defined as limits in ∗– moments of certain upper triangular random matrices. The DT–o ..."
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Cited by 26 (12 self)
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The DT–operators are introduced, one for every pair (µ, c) consisting of a compactly supported Borel probability measure µ on the complex plane and a constant c> 0. These are operators on Hilbert space that are defined as limits in ∗– moments of certain upper triangular random matrices. The DT–operators include Voiculescu’s circular operator and elliptic deformations of it, as well as the circular free Poisson operators. We show that every DT–operator is strongly decomposable. We also show that a DT–operator generates a II1–factor, whose isomorphism class depends only on the number and sizes of atoms of µ. Those DT–operators that are also R–diagonal are identified. For a quasi–nilpotent DT–operator T, we find the