We establish factoriality and non-injectivity in full generality for the mixed q-Araki–Woods von Neumann algebra associated to a separable real Hilbert space with , a strongly continuous one parameter group of orthogonal transformations on , a direct sum decomposition , and a real symmetric matrix (qij) with supi,j|qij| < 1. This is achieved by first proving the existence of conjugate variables for a finite number of generators of the algebras (when ), following the lines of Miyagawa–Speicher and Kumar–Skalski–Wasilewski. The conjugate variables belong to the factors in question and satisfy certain Lipschitz conditions.
REFERENCES
1.
R.
Speicher
, “Generalized statistics of macroscopic fields
,” Lett. Math. Phys.
27
(2
), 97
–104
(1993
).2.
M.
Bożejko
and R.
Speicher
, “Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces
,” Math. Ann.
300
, 97
–120
(1994
).3.
M.
Bożejko
and R.
Speicher
, “An example of a generalized Brownian motion
,” Commun. Math. Phys.
137
(3
), 519
–531
(1991
).4.
É.
Ricard
, “Factoriality of q-Gaussian von Neumann algebras
,” Commun. Math. Phys.
257
(3
), 659
–665
(2005
).5.
M.
Bożejko
, B.
Kümmerer
, and R.
Speicher
, “q-Gaussian processes: Non-commutative and classical aspects
,” Commun. Math. Phys.
185
(1
), 129
–154
(1997
).6.
I.
Królak
, “Factoriality of von Neumann algebras connected with general commutation relations–finite dimensional case
,” Quantum Probability
(Banach Center Publications
, 2006
), Vol. 73
, pp. 277
–284
.7.
P.
Śniady
, “Factoriality of Bożejko–Speicher von Neumann algebras
,” Commun. Math. Phys.
246
(3
), 561
–567
(2004
).8.
A.
Skalski
and S.
Wang
, “Remarks on factoriality and q-deformations
,” Proc. Am. Math. Soc.
146
(9
), 3813
–3823
(2018
).9.
I.
Królak
, “Wick product for commutation relations connected with Yang–Baxter operators and new constructions of factors
,” Commun. Math. Phys.
210
(3
), 685
–701
(2000
).10.
B.
Nelson
and Q.
Zeng
, “An application of free transport to mixed q-Gaussian algebras
,” Proc. Am. Math. Soc.
144
(10
), 4357
–4366
(2016
).11.
D.
Shlyakhtenko
, “Free quasi-free states
,” Pac. J. Math.
177
(2
), 329
–368
(1997
).12.
F.
Hiai
, q-deformed Araki-Woods algebras
, in “Operator Algebras and Mathematical Physics
(Constanta
, 2001
), pp. 169
–202
, Theta, Bucharest, 2003.13.
B.
Nelson
, “Free monotone transport without a trace
,” Commun. Math. Phys.
334
(3
), 1245
–1298
(2015
).14.
P.
Bikram
and K.
Mukherjee
, “Generator masas in q-deformed Araki–Woods von Neumann algebras and factoriality
,” J. Funct. Anal.
273
(4
), 1443
–1478
(2017
).15.
P.
Bikram
, K.
Mukherjee
, É.
Ricard
, and S.
Wang
, “On the factoriality of q-deformed Araki–Woods von Neumann algebras
,” Commun. Math. Phys.
398
(2
), 797
–821
(2023
).16.
M.
Kumar
, A.
Skalski
, and M.
Wasilewski
, “Full solution of the factoriality question for q-Araki–Woods von Neumann algebras via conjugate variables
,” Commun. Math. Phys.
402
, 157
–167
(2023
).17.
P.
Bikram
, R.
Kumar R.
, and K.
Mukherjee
, “Mixed q-deformed Araki–Woods von Neumann algebras
,” J. Noncommut. Geom.
(to be published).18.
A.
Miyagawa
and R.
Speicher
, “A dual and conjugate system for q-Gaussians for all q
,” Adv. Math.
413
, 108834
(2023
).19.
B.
Nelson
, “On finite free Fisher information for eigenvectors of a modular operator
,” J. Funct. Anal.
273
(7
), 2292
–2352
(2017
).20.
Y.
Dabrowski
and A.
Ioana
, “Unbounded derivations, free dilations, and indecomposability results for II1 factors
,” Trans. Am. Math. Soc.
368
(7
), 4525
–4560
(2015
).21.
P.
Bikram
, R.
Kumar R.
, and K.
Mukherjee
, “On non-injectivity of mixed q-deformed Araki–Woods von Neumann algebras
,” Kyoto J. Math.
(to be published).22.
A.
Nou
, “Non injectivity of the q-deformed von Neumann algebra
,” Math. Ann.
330
(1
), 17
–38
(2004
).23.
M.
Bożejko
, “Completely positive maps on Coxeter groups and the ultracontractivity of the q-Ornstein-Uhlenbeck semigroup
,” in Quantum Probability: Proceedings of the Workshop held in Gdańsk
, July 1–6, 1997, edited by R.
Alicki
, M.
Bożejko
, and W. A.
Majewski
(Banach Center Publications
, 1998
), Vol. 43
, pp. 87
–93
.24.
Z.
Yang
, “A conjugate system for twisted Araki-Woods von Neumann algebras of finite dimensional spaces
,” arXiv:2304.13856 (2023
).25.
A.
Connes
, “Une classification des facteurs de type
,” Ann. Sci. Ec. Norm. Super.
6
, 133
–252
(1973
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
You do not currently have access to this content.
