An example of a generalized Brownian motion

Bożejko, Marek; Speicher, Roland

doi:10.1007/bf02100275

Cited by 317 publications

(368 citation statements)

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“…Let H R be a real Hilbert space and H C its complexification. For −1 ≤ q ≤ 1 let F q (H C ) be the q-Fock space based on H C constructed by Bożejko and Speicher (see [BS1] and [BS2]). Note that F 1 (H C ), F −1 (H C ) and F 0 (H C ) are respectively the symmetric, anti-symmetric and full (= free) Fock spaces.…”

Section: Maximal Ergodic Inequalitiesmentioning

confidence: 99%

Noncommutative maximal ergodic theorems

Junge¹,

Xu²

2006

J. Amer. Math. Soc.

173

245

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This paper is devoted to the study of various maximal ergodic theorems in noncommutative L p L_p -spaces. In particular, we prove the noncommutative analogue of the classical Dunford-Schwartz maximal ergodic inequality for positive contractions on L p L_p and the analogue of Stein’s maximal inequality for symmetric positive contractions. We also obtain the corresponding individual ergodic theorems. We apply these results to a family of natural examples which frequently appear in von Neumann algebra theory and in quantum probability.

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Section: Maximal Ergodic Inequalitiesmentioning

confidence: 99%

Noncommutative maximal ergodic theorems

Junge¹,

Xu²

2006

J. Amer. Math. Soc.

173

245

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“…The q-relations have been studied earlier in connection with quantum fields [1] and statistical mechanics [12]. In particular, the paper [12] serves to show that the q-relations interpolate between the bosons and the fermions.…”

Section: Discussionmentioning

confidence: 99%

“…This is a difficult problem: As noted in [8], the C * -algebra A(q) on the q-relations is infinite, and its equivalence classes of irreducible representations do not admit a Borel cross section for its classification parameters. In [8], the authors showed that there is a stability interval J in the q-variable such that all the C * -algebra A(q) for q ∈ J are isomorphic to the Cuntz algebra, see [1], and they estimated the size of J.…”

Section: Introductionmentioning

confidence: 99%

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Wavelets in mathematical physics:q-oscillators

Jørgensen¹,

Paolucci²

2003

J. Phys. A: Math. Gen.

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We construct representations of a q-oscillator algebra by operators on Fock space on positive matrices. They emerge from a multiresolution scaling construction used in wavelet analysis. The representations of the Cuntz Algebra arising from this multiresolution analysis are contained as a special case in the Fock Space construction.In this paper we establish a connection between multiresolution wavelet analysis on one hand and representation theory for operator on Hilbert spaces depending on a real parameter on the other. These operators arise from a multiresolution wavelet analysis based on Bessel functions. We wish to develop a framework for the study of creation operators on Hilbert space, satisfying simple identities, and allowing a Hopf algebra structure. Examples will include oscillator algebras coming from physical models.In the first section of the paper, we review the background and the motivation for the study of the q-relations, both as it relates to problems in 1 mathematics and in physics. On the mathematical side, the problems concern wavelet analysis and transform theory, especially the Mellin transform, and on the physics side, they relate to the quon gas of statistical mechanics. For the construction of the representations, we then turn to the twisted Fock space and the q-oscillator algebra. Our approach is motivated by wavelet analysis, and it uses a certain loop group. Our main result is Theorem 6.

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On the classification and modular extendability of E₀‐semigroups on factors

Bikram

Markiewicz

2020

Mathematische Nachrichten

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In this paper we study modular extendability and equimodularity of endomorphisms and E0‐semigroups on factors, when these definitions are recast to the context of faithful normal semifinite weights and this dependency is analyzed. We show that modular extendability is a property that does not depend on the choice of weights, it is a cocycle conjugacy invariant and it is preserved under tensoring. Furthermore, we prove a necessary and sufficient condition for equimodularity of endomorphisms in the context of weights. This extends previously known results regarding the necessity of this condition in the case of states. The classification of E0‐semigroups on factors is considered: a modularly extendable E0‐semigroup is said to be of type EI, EII or EIII if its modular extension is of type I, II or III, respectively. We prove that all types exist on properly infinite factors. We show that q‐CCR flows are not extendable, and we extend previous results by the first author regarding the non‐extendability of CAR flow to a larger class of quasi‐free states. We also compute the coupling index and the relative commutant index for the CAR flows and q‐CCR flows. As an application, by considering repeated tensors of the CAR flows we show that there are infinitely many non cocycle conjugate non‐extendable E0‐semigroups on the hyperfinite factors of types II1 and IIIλ, for .

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An example of a generalized Brownian motion

Cited by 317 publications

References 24 publications

Noncommutative maximal ergodic theorems

Noncommutative maximal ergodic theorems

Wavelets in mathematical physics:q-oscillators

On the classification and modular extendability of E₀‐semigroups on factors

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An example of a generalized Brownian motion

Cited by 317 publications

References 24 publications

Noncommutative maximal ergodic theorems

Noncommutative maximal ergodic theorems

Wavelets in mathematical physics:q-oscillators

On the classification and modular extendability of E0‐semigroups on factors

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Product

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On the classification and modular extendability of E₀‐semigroups on factors