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## Banach Center Publications

1998 | 43 | 1 | 9-24
Tytuł artykułu

### Singleton independence

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
9-24
Opis fizyczny
Daty
wydano
1998
Twórcy
autor
• Centro Vito Volterra, Università di Roma "Tor Vergata", 00133 Roma, Italy
autor
• Graduate School of Polymathematics, Nagoya University, Nagoya 464-8602, Japan
autor
• Graduate School of Polymathematics, Nagoya University, Nagoya 464-8602, Japan
Bibliografia
• [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.
• [2] L. Accardi, A. Frigerio and J. Lewis, Quantum stochastic processes, Publ. RIMS Kyoto University 18 (1982), 97-133.
• [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.
• [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.
• [5] M. Bożejko, Uniformly bounded representations of free groups, J. Reine Angew. Math. 377 (1987), 170-186.
• [6] M. Bożejko, Positive definite kernels, length functions on groups and noncommutative von Neumann inequality, Studia Math. 95 (1989), 107-118.
• [7] M. Bożejko, Harmonic analysis on discrete groups and noncommutative probability, Volterra preprint series No. 93, 1992.
• [8] M. Bożejko, private communication, November, 1997.
• [9] M. Bożejko, B. Kümmerer and R. Speicher, q-Gaussian processes: Non-commutative and classical aspects, Commun. Math. Phys. 185 (1997), 129-154.
• [10] M. Bożejko, M. Leinert and R. Speicher, Convolution and limit theorems for conditionally free random variables, Pacific J. Math. 175 (1996), 357-388.
• [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.
• [12] I. Chiswell, Abstract length functions in groups, Math. Proc. Camb. Phil. Soc. 80 (1976), 451-463.
• [13] F. Fagnola, A Lévy theorem for free noises, Probab. Th. Rel. Fields 90 (1991), 491-504. %Preprint,1991, Rendiconti Accademia dei Lincei (1992).
• [14] A. Figà-Talamanca and M. Picardello, Harmonic Analysis on Free Groups, Marcel Dekker, New York and Basel, 1983.
• [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.
• [16] N. Giri and W. von Waldenfels, An algebraic version of the central limit theorem, ZW 42 (1978), 129-134.
• [17] U. Haagerup, An example of a non-nuclear C*-algebra which has the metric approximation property, Invent. Math. 50 (1979), 279-293.
• [18] Y. Hashimoto, Deformations of the semi-circle law derived from random walks on free groups, to appear in Prob. Math. Stat. 18 (1998).
• [19] F. Hiai and D. Petz, Maximizing free entropy, Preprint No.17, Mathematical Institute, Hungarian Academy of Sciences, Budapest, 1996.
• [20] A. Hora, Central limit theorems and asymptotic spectral analysis on large graphs, submitted to Infinite Dimensional Analysis and Quantum Probability, 1997.
• [21] R. Lenczewski, Quantum central limit theorems, in: Symmetries in Sciences VIII (B. Gruber, ed.), pp. 299-314, Plenum, 1995.
• [22] V. Liebscher, Note on entangled ergodic theorems, preprint, 1997.
• [23] R. Lyndon, Length functions in groups, Math. Scand. 12 (1963), 209-234.
• [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.
• [25] M. Schürmann, White Noise on Bialgebras, Lect. Notes in Math. Vol. 1544, Springer-Verlag, 1993.
• [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.
• [27] D. Voiculescu, Free noncommutative random variables, random matrices and the $II_1$ factors of free groups, in: Quantum Probability and Related Fields VI, pp. 473-487, World Scientific, 1991.
• [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.
• [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.
Typ dokumentu
Bibliografia
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