We will investigate the α-z-Rényi divergence in the general von Neumann algebra setting based on Haagerup non-commutative Lp-spaces. In particular, we establish almost all its expected properties when 0 < α < 1 and some of them when α > 1. In an Appendix we also give an equality condition for generalized Hölder’s inequality in Haagerup non-commutative Lp-spaces.
REFERENCES
1.
H.
Umegaki
, “Conditional expectation in an operator algebra. IV. Entropy and information
,” Kodai Math. J.
14
, 59
(1962
).2.
D.
Petz
, Rep. Math. Phys.
23
, 57
(1986
).3.
M.
Müller-Lennert
, F.
Dupuis
, O.
Szehr
, S.
Fehr
, and M.
Tomamichel
, J. Math. Phys.
54
, 122203
(2013
).4.
M. M.
Wilde
, A.
Winter
, and D.
Yang
, Commun. Math. Phys.
331
, 593
(2014
).5.
S.
Beigi
, J. Math. Phys.
54
, 122202
–122211
(2013
).6.
R. L.
Frank
and E. H.
Lieb
, J. Math. Phys.
54
, 122201
(2013
).7.
K. M. R.
Audenaert
and N.
Datta
, J. Math. Phys.
56
, 022202
(2015
).8.
V.
Jakšić
, Y.
Ogata
, Y.
Pautrat
, and C.-A.
Pillet
, Quantum Theory from Small to Large Scales: Lecture Notes of the Les Houches Summer School
(Oxford University Press
, 2012
), Vol. 95
.9.
E. A.
Carlen
, R. L.
Frank
, and E. H.
Lieb
, J. Phys. A: Math. Theor.
51
, 483001
(2018
).10.
H.
Zhang
, Adv. Math.
365
, 107053
(2020
).11.
H.
Araki
, Publ. Res. Inst. Math. Sci.
11
, 809
(1975
).12.
H.
Araki
, Publ. Res. Inst. Math. Sci.
13
, 173
(1977
).13.
F.
Hiai
, J. Math. Phys.
59
, 102202
–102227
(2018
).14.
D.
Petz
, Publ. Res. Inst. Math. Sci.
21
, 787
(1985
).15.
M.
Berta
, V. B.
Scholz
, and M.
Tomamichel
, Ann. Henri Poincaré
19
, 1843
(2018
).16.
A.
Jenčová
, Ann. Henri Poincaré
19
, 2513
(2018
).17.
A.
Jenčová
, Ann. Henri Poincaré
22
, 3235
(2021
).18.
F.
Hiai
, Quantum f-divergences in von Neumann algebras—Reversibility of quantum operations
, Mathematical Physics Studies
(Springer
, Singapore
, 2021
), p. x+194
.19.
M.
Mosonyi
, Commun. Math. Phys.
400
, 83
(2023
).20.
T.
Zhang
, X.
Qi
, and J.
Banach
, Banach J. Math. Anal.
17
, 22
(2023
).21.
S.
Kato
and Y.
Ueda
, “A remark on non-commutative Lp-spaces
,” arXiv:2307.01790 [math.OA] (2023
), Studia Math., to appear.22.
M.
Terp
, “Lp spaces associated with von Neumann algebras. notes
,” Copenhagen University Lecture Notes, unpublished.23.
F.
Hiai
, Lectures on selected topics in von Neumann algebras
, EMS Series of Lectures in Mathematics
(EMS Press
, Berlin
, 2021
), p. viii+241
.24.
T.
Fack
and H.
Kosaki
, Pacific J. Math.
123
, 269
(1986
).25.
F.
Hiai
and Y.
Nakamura
, Pacific J. Math.
138
, 259
(1989
).26.
H.
Kosaki
, Trans. Am. Math. Soc.
283
, 265
(1984
).27.
F.
Hiai
and M.
Mosonyi
, Ann. Henri Poincaré
24
, 1681
(2023
).28.
M. D.
Choi
, Illinois J. Math.
18
, 565
(1974
).29.
H.
Kosaki
, Proc. Am. Math. Soc.
114
, 477
(1992
).31.
J. B.
Conway
, A Course in Functional Analysis
, Graduate Texts in Mathematics
, 2nd ed. (Springer-Verlag
, New York
, 1990
), Vol. 96
, p. xvi+399
.32.
Ş.
Strătilă
, Modular Theory in Operator Algebras
(Editura Academiei Republicii Socialiste România, Bucharest; Abacus Press
, Tunbridge Wells
, 1981
), p. 492
, translated from the Romanian by the author.33.
E. A.
Carlen
and E. H.
Lieb
, Lett. Math. Phys.
83
, 107
(2008
).34.
E. A.
Carlen
and H.
Zhang
, Linear Algebra Appl.
654
, 289
(2022
).35.
G.
Larotonda
, Math. Proc. R. Irish Acad.
118A
, 1
(2018
).© 2024 Author(s). Published under an exclusive license by AIP Publishing.
2024
Author(s)
You do not currently have access to this content.
