No Access

THE FIVE INDEPENDENCES AS NATURAL PRODUCTS

NAOFUMI MURAKI

Mathematics Laboratory, Iwate Prefectural University, Takizawa, Iwate 020-0193, Japan

https://doi.org/10.1142/S0219025703001365Cited by:53 (Source: Crossref)

Abstract

Let be the class of all algebraic probability spaces. A "natural product" is, by definition, a map which is required to satisfy all the canonical axioms of Ben Ghorbal and Schürmann for "universal product" except for the commutativity axiom. We show that there exist only five natural products, namely tensor product, free product, Boolean product, monotone product and anti-monotone product. This means that, in a sense, there exist only five universal notions of stochastic independence in noncommutative probability theory.

Keywords:

References

L. Accardi , Y. G. Lu and I. Volovich , Quantum Theory and its Stochastic Limit ( Springer , 2002 ) . Crossref, Google Scholar
A. Ben Ghorbal and M. Schürmann, Math. Proc. Camb. Phil. Soc. 133, 531 (2002). Crossref, Web of Science, Google Scholar
M. Bożejko, J. Reine Angew. Math. 377, 170 (1987). Web of Science, Google Scholar
D. D. Cushen and R. L. Hudson, J. Appl. Probab. 8, 454 (1971), DOI: 10.2307/3212170. Crossref, Web of Science, Google Scholar
U. Franz, Infinite Dim. Anal. Quantum Probab. 4, 401 (2001), DOI: 10.1142/S0219025701000565. Link, Web of Science, Google Scholar
U. Franz, Unification of boolean, monotone, anti-monotone, and tensor independence and Lévy processes, preprint, EMAU Greifswald, 2001. To appear in Math. Z. . Google Scholar
N. Giri and W. von Waldenfels, Z. Wahr. Verw. Gebiete 42, 129 (1978), DOI: 10.1007/BF00536048. Crossref, Google Scholar
F. Hiai and D. Petz , The Semicircle Law, Free Random Variables and Entropy , Mathematical Surveys and Monographs 77 ( Amer. Math. Soc. , 2000 ) . Google Scholar
R. L. Hudson, J. Appl. Probab. 10, 502 (1973), DOI: 10.2307/3212771. Crossref, Web of Science, Google Scholar
Y. G. Lu, Probab. Math. Statist. 17, 149 (1997). Web of Science, Google Scholar
P. A. Meyer , Quantum Probability for Probabilists , Lecture Notes in Math. 1538 ( Springer , 1993 ) . Crossref, Google Scholar
N. Muraki, A new example of noncommutative "de Moivre-Laplace theorem", Probability Theory and Mathematical Statistics, Proc. Seventh Japan–Russia Symp., eds. S. Watanabeet al. (World Scientific, 1996) pp. 353–362. Google Scholar
N. Muraki, Commun. Math. Phys. 183, 557 (1997), DOI: 10.1007/s002200050043. Crossref, Web of Science, Google Scholar
N. Muraki, Infinite Dim. Anal. Quantum Probab. 4, 39 (2001), DOI: 10.1142/S0219025701000334. Link, Web of Science, Google Scholar
N. Muraki, Monotonic convolution and monotonic Lévy–Hinčin formula, preprint, Iwate Pref. Univ., 2000, submitted to Probab. Theory Rel. Fields . Google Scholar
N. Muraki, Infinite Dim. Anal. Quantum Probab. 5, 113 (2002), DOI: 10.1142/S0219025702000742. Link, Web of Science, Google Scholar
K. R. Parthasarathy , An Introduction to Quantum Stochastic Calculus ( Birkhäuser , 1992 ) . Crossref, Google Scholar
M. Schürmann , White Noise on Bialgebras , Lecture Notes in Math. 1544 ( Springer , 1993 ) . Crossref, Google Scholar
M. Schürmann, J. Funct. Anal. 133, 1 (1995), DOI: 10.1006/jfan.1995.1114. Crossref, Web of Science, Google Scholar
M. Schürmann, Non-commutative probability on algebraic structures, Probability Measures on Groups and Related Structures XI, Proc., ed. H. Heyer (World Scientific, 1995) pp. 332–356. Google Scholar
R. Speicher, Probab. Theorem Rel. Fields 84, 141 (1990), DOI: 10.1007/BF01197843. Crossref, Web of Science, Google Scholar
R. Speicher, On universal products, Free Probability Theory, Proc.12, Fields Inst. Commun., ed. D. V. Voiculescu (Amer. Math. Soc., 1997) pp. 257–266. Google Scholar
R. Speicher and R. Woroudi, Boolean convolution, Free Probability Theory, Proc.12, Fields Inst. Commun., ed. D. V. Voiculescu (Amer. Math. Soc., 1997) pp. 267–279. Google Scholar
D. V. Voiculescu , K. J. Dykema and A. Nica , Free Random Variables , CRM Monograph Series ( Amer. Math. Soc. , 1992 ) . Crossref, Google Scholar
D. V. Voiculescu, Symmetries of some reduced free product C*-algebras, Operator Algebras and their Connections with Topology and Ergodic Theory, Proc.1132, Lecture Notes in Math., eds. H. Arakiet al. (Springer, 1985) pp. 556–588. Google Scholar
D. V. Voiculescu, J. Operator Theory 17, 85 (1987). Web of Science, Google Scholar
W. von Waldenfels, Probability and Information Theory II, Lecture Notes in Math. 296, eds. M. Behara, K. Krickeberg and J. Wolfowitz (Springer, 1973) pp. 19–69. Crossref, Google Scholar
W. von Waldenfels, Z. Wahr. Verw. Gebiete 42, 135 (1978). Crossref, Google Scholar