## The five independences as natural products.(English)Zbl 1053.81057

This paper is devoted to the study of noncommutative notions of stochastic independence in quantum probability theory or noncommutative probability theory. There are three fundamental notions of independence: namely, tensor independence, cf. W. von Waldenfels [Z. Wahrsch, Verw. Geb. 42, 135–140 (1978; Zbl 0405.60095)], free independence, cf. D. V. Voiculescu, K. J. Dykema and A. Nica [Free Random Variables, CRM Monograph Series 1, Providence, Am. Math. Soc. (AMS), v, 70p. (1992; Zbl 0795.46049)], and Boolean independence, cf. R. Speicher and R. Woroudi [Voiculescu, Dan-Virgil (ed.), Free probability theory. Papers from a workshop on random matrices and operator algebra free products, Toronto, Canada, Mars 1995. Providence, RI: American Mathematical Society. Fields Inst. Commun. 12, 267–279 (1997; Zbl 0872.46033)]. These three independences are the only possible universal notions in Schürmann’s sense, see [M. Schürmann, J. Funct. Anal. 133, No.1, 1–9 (1995; Zbl 0868.46048)]. The author recently found anather example of independence named “monotone independence” for $$C^*$$-probability spaces, which possesses a certain universal character except the commutativity axiom, cf. N. Muraki [Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4, No. 1, 39–58 (2001; Zbl 1046.46049)], see also U. Franz [Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4, No.3, 401–407 (2001; Zbl 1039.81035)]. In the author’s previous work [Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5, No.1, 113–134 (2002; Zbl 1055.46514)], it is proved that the known five products (tensor, free, Boolean, monotone and anti-monotone) are the only possible quasi-universal products. This paper gives a complete positive answer to the expectation in the canonical setting of Ben Ghorbal and Schürmann that the above five indendences are the only possible universal notions of stochastic independence in noncommutative probability theory.

### MSC:

 81S25 Quantum stochastic calculus 46L54 Free probability and free operator algebras 60A05 Axioms; other general questions in probability
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### References:

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