A simple proof of the classification theorem for positive natural products

• Autorzy: Muraki N.
• Języki publikacji: EN

Abstrakty

EN

A simplification of the proof of the classification theorem for natural notions of stochastic independence is given. This simplification is made possible after adding the positivity condition to the algebraic axioms for a (non-symmetric) universal product (i.e. a natural product). Indeed, this simplification is nothing but a simplification, under the positivity, of the proof of the claim that, for any natural product, the ‘wrong-ordered’ coefficients all vanish in the expansion form. The known proof of this claim involves a cumbersome process of solving a system of quadratic equations in 102 unknowns, but in our new proof under the positivity we can avoid such a process.

Słowa kluczowe

EN

non-commutative probability; quantum probability; independence; universal products; natural products

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2013
• Tom: Vol. 33, Fasc. 2
• Strony: 315--326
• Opis fizyczny: Bibliogr. 6 poz.

Twórcy

autor

Muraki N.

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

Bibliografia

• [1] A. Ben Ghorbal and M. Schürmann, Non-commutative notions of stochastic independence, Math. Proc. Cambridge Philos. Soc. 133 (2002), pp. 531-561.
• [2] A. Ben Ghorbal and M. Schürmann, Quantum Lévy processes on dual groups, Math. Z. 251 (2005), pp. 147-165.
• [3] N. Muraki, The five independences as quasi-universal products, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 (2002), pp. 113-134.
• [4] N. Muraki, The five independences as natural products, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), pp. 337-371.
• [5] M. Schürmann, Direct sums of tensor products and non-commutative independence, J. Funct. Anal. 133 (1995), pp. 1-9.
• [6] R. Speicher, On universal products, in: Free Probability Theory, D. Voiculescu (Ed.), Fields Inst. Commun., Vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 257-266.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).