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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 120 (7 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].
On ergodic theorems for free group actions on noncommutative spaces
, 2005
"... Abstract. We extend in a noncommutative setting the individual ergodic theorem of Nevo and Stein concerning measure preserving actions of free groups and averages on spheres s2n of even radius. Here we study state preserving actions of free groups on a von Neumann algebra A and the behaviour of (s2n ..."
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Abstract. We extend in a noncommutative setting the individual ergodic theorem of Nevo and Stein concerning measure preserving actions of free groups and averages on spheres s2n of even radius. Here we study state preserving actions of free groups on a von Neumann algebra A and the behaviour of (s2n(x)) for x in noncommutative spaces Lp (A). For ∑ n−1 k=0 sk and p = +∞, this problem was solved by the Cesàro means 1 n Walker. Our approach is based on ideas of Bufetov. We prove a noncommutative version of Rota “Alternierende Verfahren ” theorem. To this end, we introduce specific dilations of the powers of some noncommutative Markov operators. 1.
Quantum filter processes driven by Markovian white noises have classical versions
, 903
"... We study quantum filters that are driven by basic quantum noises and construct classical versions. Our approach is based on exploiting the quantum markovian component of the observation and measurement processes of the filters. This approach leads in a natural way the classical versions for a class ..."
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We study quantum filters that are driven by basic quantum noises and construct classical versions. Our approach is based on exploiting the quantum markovian component of the observation and measurement processes of the filters. This approach leads in a natural way the classical versions for a class of quantum filters. We consider quantum white noises derived from Wiener and Poisson processes that drive the signal and measurement processes and derive the recursive filtering equations using classical machinery. 1
RECONSTRUCTION THEOREM FOR STATIONARY MONOTONE QUANTUM MARKOV PROCESSES
"... Abstract. Based on the Hilbert Cmodule structure we study the reconstruction theorem for stationary monotone quantumMarkov processes from quantum dynamical semigroups. We prove that the quantum stochastic monotone process constructed from a covariant quantum dynamical semigroup is again covarian ..."
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Abstract. Based on the Hilbert Cmodule structure we study the reconstruction theorem for stationary monotone quantumMarkov processes from quantum dynamical semigroups. We prove that the quantum stochastic monotone process constructed from a covariant quantum dynamical semigroup is again covariant in the strong sense. 1.