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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 73 (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].
Free Probability Theory And Free Diffusion
"... Introduction Free probability theory was introduced and developed by Dan Voiculescu in an operator algebraic context, but has since then turned out to possess links to a lot of quite dierent elds of mathematics and physics. I will give a short general introduction into the basics of free probabilit ..."
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Introduction Free probability theory was introduced and developed by Dan Voiculescu in an operator algebraic context, but has since then turned out to possess links to a lot of quite dierent elds of mathematics and physics. I will give a short general introduction into the basics of free probability and illuminate certain aspects of that theory (in particular, the analogy between classical and free probability theory) by a closer look at free diusion. An extensive presentation of the basic theory of free probability is given in the monograph [VDN], whereas for getting an impression of the diversity of this eld one should consult [V2, V3]. 2. Free probability theory Free probability theory was introduced by Dan Voiculescu around 1985 as a tool for investigating the structure of special von Neumann algebras. Voiculescu separated from that concrete context the following abstract concept of 'freeness' and found it worth to be investigated on its own sake. The
FREE CALCULUS
, 2001
"... . Let me first recall the basic definitions and some fundamental realizations of freeness. For a more extensive review, I refer to the course of Biane or to the books [18, 17] (see also the survey [16]). 1.1. Definitions. 1) A (noncommutative) probability space consists of a pair (A, ϕ), where • A ..."
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. Let me first recall the basic definitions and some fundamental realizations of freeness. For a more extensive review, I refer to the course of Biane or to the books [18, 17] (see also the survey [16]). 1.1. Definitions. 1) A (noncommutative) probability space consists of a pair (A, ϕ), where • A is a unital algebra • ϕ: A → C is a unital linear functional, i.e. in particular ϕ(1) = 1 2) Unital subalgebras A1,..., An ⊂ A are called free, if we have whenever ai ∈ Aj(i) ϕ(a1...ak) = 0 (i = 1,...,k) j(1) ̸ = j(2) ̸ = · · · ̸ = j(k) ϕ(ai) = 0 (i = 1,..., k) 3) Random variables x1,...,xn ∈ A are called free, if A1,...,An are free, where Ai is the unital algebra generated by xi.. A canonical realization of free random variables is given on the full Fock space. 1.2. Definitions. Let H be a Hilbert space. 1) The full Fock space over H is the Hilbert space