The analogues of entropy and of Fisher's information measure in free probability theory. V: Noncommutative Hilbert transforms
type: Item
instance of: scholarly article
title: The analogues of entropy and of Fisher's information measure in free probability theory. V: Noncommutative Hilbert transforms
zbMATH Open document ID: 0930.46053
published in: Inventiones Mathematicae
publication date: 2000-02-15T00:00:00Z
full work available at URL: https://eudml.org/doc/144242
review text: [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)).
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 94A17
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 60A99
zbMATH DE Number: 1186160
zbMATH Keywords: noncommutative probability
zbMATH Keywords: free Fisher information measure
zbMATH Keywords: free entropy
zbMATH Keywords: noncommutative generalization of the Hilbert transform
author: Dan Voiculescu
reviewed by: A. F. Gualtierotti
MaRDI profile type: Publication
OpenAlex ID: W2043256123
DOI: 10.1007/S002220050222
description: scientific article; zbMATH DE number 1186160