Results 1  10
of
134
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 ..."
Abstract

Cited by 120 (7 self)
 Add to MetaCart
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].
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 ..."
Abstract

Cited by 87 (1 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 81 (3 self)
 Add to MetaCart
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 products of hyperfinite von Neumann algebras and free dimension
 DUKE MATH. J
, 1992
"... ..."
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 ..."
Abstract

Cited by 60 (4 self)
 Add to MetaCart
(Show Context)
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
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 ..."
Abstract

Cited by 59 (9 self)
 Add to MetaCart
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 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 ..."
Abstract

Cited by 57 (4 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 53 (8 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 48 (14 self)
 Add to MetaCart
(Show Context)
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
Khintchine type inequalities for reduced free products and applications
, 2005
"... We prove Khintchine type inequalities for words of a fixed length in a reduced free product of C ∗algebras (or von Neumann algebras). These inequalities imply that the natural projection from a reduced free product onto the subspace generated by the words of a fixed length d is completely bounded w ..."
Abstract

Cited by 36 (3 self)
 Add to MetaCart
(Show Context)
We prove Khintchine type inequalities for words of a fixed length in a reduced free product of C ∗algebras (or von Neumann algebras). These inequalities imply that the natural projection from a reduced free product onto the subspace generated by the words of a fixed length d is completely bounded with norm depending linearly on d. We then apply these results to various approximation properties on reduced free products. As a first application, we give a quick proof of Dykema’s theorem on the stability of exactness under the reduced free product for C ∗algebras. We next study the stability of the completely contractive approximation property (CCAP) under reduced free product. Our first result in this direction is that a reduced free product of finite dimensional C ∗algebras has the CCAP. The second one asserts that a von Neumann reduced free product of injective von Neumann algebras has the weak ∗ CCAP. In the case of group C ∗algebras, we show that a free product of weakly amenable groups with constant 1 is weakly amenable.