Motivated by the central limit problem for algebraic probability spaces arising from the Haagerup states on the free group with countably infinite generators, we introduce a new notion of statistical independence in terms of inequalities rather than of usual algebraic identities. In the case of the Haagerup states the role of the Gaussian law is played by the Ullman distribution. The limit process is realized explicitly on the finite temperature Boltzmannian Fock space. Furthermore, a functional central limit theorem associated with the Haagerup states is proved and the limit white noise is investigated.
@article{bwmeta1.element.bwnjournal-article-bcpv43i1p9bwm,
author = {Accardi, Luigi and Hashimoto, Yukihiro and Obata, Nobuaki},
title = {Singleton independence},
journal = {Banach Center Publications},
volume = {43},
year = {1998},
pages = {9-24},
zbl = {0929.60002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv43i1p9bwm}
}
Accardi, Luigi; Hashimoto, Yukihiro; Obata, Nobuaki. Singleton independence. Banach Center Publications, Tome 43 (1998) pp. 9-24. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv43i1p9bwm/
[000] [1] L. Accardi, I. Ya. Aref'eva and I. V. Volovich, The master field for half-planar diagrams and free non-commutative random variables, to appear in Quarks `96 (V. Matveev and V. Rubakov, eds.), HEP-TH/9502092.
[001] [2] L. Accardi, A. Frigerio and J. Lewis, Quantum stochastic processes, Publ. RIMS Kyoto University 18 (1982), 97-133. | Zbl 0498.60099
[002] [3] L. Accardi, S. V. Kozyrev and I. V. Volovich, Dynamics of dissipative two-state systems in the stochastic approximation, Phys. Rev. A 56 (1997), 1-7.
[003] [4] L. Accardi, Y. Hashimoto and N. Obata, Notions of independence related to the free group, Infinite Dimen. Anal. Quantum Probab. 1 (1998), 221-246. | Zbl 0913.46057
[004] [5] M. Bożejko, Uniformly bounded representations of free groups, J. Reine Angew. Math. 377 (1987), 170-186. | Zbl 0604.43004
[005] [6] M. Bożejko, Positive definite kernels, length functions on groups and noncommutative von Neumann inequality, Studia Math. 95 (1989), 107-118. | Zbl 0714.43007
[006] [7] M. Bożejko, Harmonic analysis on discrete groups and noncommutative probability, Volterra preprint series No. 93, 1992.
[007] [8] M. Bożejko, private communication, November, 1997.
[008] [9] M. Bożejko, B. Kümmerer and R. Speicher, q-Gaussian processes: Non-commutative and classical aspects, Commun. Math. Phys. 185 (1997), 129-154.
[009] [10] M. Bożejko, M. Leinert and R. Speicher, Convolution and limit theorems for conditionally free random variables, Pacific J. Math. 175 (1996), 357-388. | Zbl 0874.60010
[010] [11] M. Bożejko and R. Speicher, ψ-Independent and symmetrized white noises, in: Quantum Probability and Related Fields VI, pp. 219-236, World Scientific, 1991.
[011] [12] I. Chiswell, Abstract length functions in groups, Math. Proc. Camb. Phil. Soc. 80 (1976), 451-463. | Zbl 0351.20024
[012] [13] F. Fagnola, A Lévy theorem for free noises, Probab. Th. Rel. Fields 90 (1991), 491-504. %Preprint,1991, Rendiconti Accademia dei Lincei (1992). | Zbl 0729.60074
[013] [14] A. Figà-Talamanca and M. Picardello, Harmonic Analysis on Free Groups, Marcel Dekker, New York and Basel, 1983. | Zbl 0536.43001
[014] [15] M. de Giosa and Y. G. Lu, From quantum Bernoulli process to creation and annihilation operators on interacting q-Fock space, to appear in Nagoya Math. J. | Zbl 0916.60082
[015] [16] N. Giri and W. von Waldenfels, An algebraic version of the central limit theorem, ZW 42 (1978), 129-134. | Zbl 0362.60043
[016] [17] U. Haagerup, An example of a non-nuclear C*-algebra which has the metric approximation property, Invent. Math. 50 (1979), 279-293. | Zbl 0408.46046
[017] [18] Y. Hashimoto, Deformations of the semi-circle law derived from random walks on free groups, to appear in Prob. Math. Stat. 18 (1998).
[018] [19] F. Hiai and D. Petz, Maximizing free entropy, Preprint No.17, Mathematical Institute, Hungarian Academy of Sciences, Budapest, 1996. | Zbl 0913.94003
[019] [20] A. Hora, Central limit theorems and asymptotic spectral analysis on large graphs, submitted to Infinite Dimensional Analysis and Quantum Probability, 1997.
[020] [21] R. Lenczewski, Quantum central limit theorems, in: Symmetries in Sciences VIII (B. Gruber, ed.), pp. 299-314, Plenum, 1995. | Zbl 0952.60021
[021] [22] V. Liebscher, Note on entangled ergodic theorems, preprint, 1997.
[022] [23] R. Lyndon, Length functions in groups, Math. Scand. 12 (1963), 209-234. | Zbl 0119.26402
[023] [24] N. Muraki, A new example of noncommutative 'de Moivre-Laplace theorem', in: Probability Theory and Mathematical Statistics (S. Watanabe et al., eds.), pp. 353-362, World Scientific, 1996. | Zbl 1147.81307
[024] [25] M. Schürmann, White Noise on Bialgebras, Lect. Notes in Math. Vol. 1544, Springer-Verlag, 1993. | Zbl 0773.60100
[025] [26] R. Speicher and W. von Waldenfels, A general central limit theorem and invariance principle, in: Quantum Probability and Related Topics IX, pp. 371-387, World Scientific, 1994.
[026] [27] D. Voiculescu, Free noncommutative random variables, random matrices and the factors of free groups, in: Quantum Probability and Related Fields VI, pp. 473-487, World Scientific, 1991. | Zbl 0928.46045
[027] [28] W. von Waldenfels, An approach to the theory of pressure broadening of spectral lines, in: Probability and Information Theory II (M. Behara et al, eds.), pp. 19-69, Lect. Notes in Math. Vol. 296, Springer-Verlag, 1973.
[028] [29] W. von Waldenfels, Interval partitions and pair interactions, in: Séminaire de Probabilités IX (P. A. Meyer, ed.), pp. 565-588, Lect. Notes in Math. Vol. 465, Springer-Verlag, 1975.