The analogues of entropy and of Fisher's information measure in free probability theory. V: Noncommutative Hilbert transforms
[For parts I--IV see Commun. Math. Phys. 155, 71-92 (1993; Zbl 0781.60006), Invent. Math. 118, 411-440 (1994; Zbl 0820.60001), Geom. Funct. Anal. 6, 172-199 (1996; Zbl 0856.60012), and in Free probability theory. Fields Inst. Commun. 12, 293-302 (1997), respectively.] In Part II (loc. cit.) the author produced a definition of free entropy, i.e., in his own words, ``a quantity that plays the role of entropy in noncommutative probability theory when usual independence is replaced by free independence'', that allowed him to obtain a number of satisfactory results, but not all of those that are deemed necessary. That is why a new, alternative definition is adopted and used in the paper under review. In the author's terms: ``In this new approach the free Fisher information measure comes first and the free entropy is derived from it. A noncommutative generalization of the Hilbert transform plays a key role in the (our) definition''. However, even with this new approach, not all the required results have been derived and the relationships which may exist between the two definitions remain to be established. Part VI: Liberation and mutual free information has been published (see Adv. Math. 146, 101-166 (1999)).
noncommutative generalization of the Hilbert transform
scientific article; zbMATH DE number 1186160