On the structure of non-commutative white noises

Köstler, Claus; Speicher, Roland

doi:10.1090/S0002-9947-07-04165-7

On the structure of non-commutative white noises

Authors: Claus Köstler and Roland Speicher
Journal: Trans. Amer. Math. Soc. 359 (2007), 4325-4338
MSC (2000): Primary 46L53; Secondary 46L55
DOI: https://doi.org/10.1090/S0002-9947-07-04165-7
Published electronically: April 16, 2007
MathSciNet review: 2309187
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Abstract: We consider the concepts of continuous Bernoulli systems and non-commutative white noises. We address the question of isomorphism of continuous Bernoulli systems and show that for large classes of quantum Levy processes one can make quite precise statements about the time behaviour of their moments.

[Arv] William Arveson, Noncommutative dynamics and 𝐸-semigroups, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. MR 1978577
[BKS] Marek Bożejko, Burkhard Kümmerer, and Roland Speicher, 𝑞-Gaussian processes: non-commutative and classical aspects, Comm. Math. Phys. 185 (1997), no. 1, 129–154. MR 1463036, https://doi.org/10.1007/s002200050084
[BSp1] Marek Bożejko and Roland Speicher, An example of a generalized Brownian motion, Comm. Math. Phys. 137 (1991), no. 3, 519–531. MR 1105428
[BSp2] Marek Bożejko and Roland Speicher, Interpolations between bosonic and fermionic relations given by generalized Brownian motions, Math. Z. 222 (1996), no. 1, 135–159. MR 1388006, https://doi.org/10.1007/PL00004258
[BSp3] Marek Bożejko and Roland Speicher, Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces, Math. Ann. 300 (1994), no. 1, 97–120. MR 1289833, https://doi.org/10.1007/BF01450478
[GM1] Mădălin Guţă and Hans Maassen, Symmetric Hilbert spaces arising from species of structures, Math. Z. 239 (2002), no. 3, 477–513. MR 1893849, https://doi.org/10.1007/s002090100316
[GM2] Mădălin Guţă and Hans Maassen, Generalised Brownian motion and second quantisation, J. Funct. Anal. 191 (2002), no. 2, 241–275. MR 1911186, https://doi.org/10.1006/jfan.2001.3855
[HKK] J. Hellmich, C. Köstler, and B. Kümmerer: Non-commutative continuous Bernoulli shifts. arXiv:math.OA/0411565 v1, 2004.
[JX] Marius Junge and Quanhua Xu, Noncommutative Burkholder/Rosenthal inequalities, Ann. Probab. 31 (2003), no. 2, 948–995. MR 1964955, https://doi.org/10.1214/aop/1048516542
[Kös1] C. Köstler, Quanten-Markoff-Prozesse und Quanten-Brownsche Bewegungen. Ein operatoralgebraischer Zugang. Ph.D. Thesis, Stuttgart (2000).
[Kös2] Claus Köstler, Survey on a quantum stochastic extension of Stone’s theorem, Advances in quantum dynamics (South Hadley, MA, 2002) Contemp. Math., vol. 335, Amer. Math. Soc., Providence, RI, 2003, pp. 209–222. MR 2029625, https://doi.org/10.1090/conm/335/06010
[Kös3] C. Köstler, On the relationship of continuous Bernoulli systems, Tsirelson's continuous products of probability spaces and Arveson's product systems. In preparation.
[Kös4] C. Köstler, An operator algebraic approach to quantum Lévy processes and quantum Brownian motions. Preprint.
[Kro] Ilona Królak, Contractivity properties of Ornstein-Uhlenbeck semigroup for general commutation relations, Math. Z. 250 (2005), no. 4, 915–937. MR 2180382, https://doi.org/10.1007/s00209-005-0801-1
[Küm1] Burkhard Kümmerer, Markov dilations on 𝑊*-algebras, J. Funct. Anal. 63 (1985), no. 2, 139–177. MR 803091, https://doi.org/10.1016/0022-1236(85)90084-9
[Küm2] Burkhard Kümmerer, Quantum white noise, Infinite-dimensional harmonic analysis (Tübingen, 1995) Gräbner, Tübingen, 1996, pp. 156–168. MR 1406579
[Pop] Sorin Popa, Orthogonal pairs of ∗-subalgebras in finite von Neumann algebras, J. Operator Theory 9 (1983), no. 2, 253–268. MR 703810
[PX1] Gilles Pisier and Quanhua Xu, Non-commutative martingale inequalities, Comm. Math. Phys. 189 (1997), no. 3, 667–698. MR 1482934, https://doi.org/10.1007/s002200050224
[PX2] Gilles Pisier and Quanhua Xu, Non-commutative 𝐿^{𝑝}-spaces, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1459–1517. MR 1999201, https://doi.org/10.1016/S1874-5849(03)80041-4
[SpW] R. Speicher, and W. von Waldenfels, A general central limit theorem and invariance principle. In Quantum Probability and Related Topics IX (1994), 371-387.
[Tsi] Boris Tsirelson, Nonclassical stochastic flows and continuous products, Probab. Surv. 1 (2004), 173–298. MR 2068474, https://doi.org/10.1214/154957804100000042
[Voi] Dan Voiculescu, Free entropy, Bull. London Math. Soc. 34 (2002), no. 3, 257–278. MR 1887698, https://doi.org/10.1112/S0024609301008992
[VDN] D. V. Voiculescu, K. J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. MR 1217253