We study certain non-tracial von Neumann algebras generated by some self-adjoint operators satisfying mixed q-commutation relations. Such algebras are discussed in the work of Bikram et al. [“Mixed q-deformed Araki-Woods von Neumann algebras,” (submitted)]. We prove the analog of Nelson’s hypercontractivity inequality for the mixed q-Ornstein–Uhlenbeck semigroup. We also show that the mixed q-Ornstein–Uhlenbeck semigroup is ultracontractive.
REFERENCES
1.
P.
Bikram
, R.
Kumar
, and K. K.
Mukherjee
, “Mixed q-deformed Araki-Woods von Neumann algebras
,” (submitted).2.
E.
Nelson
, “The free Markoff field
,” J. Funct. Anal.
12
, 211
–227
(1973
).3.
L.
Gross
, “Logarithmic Sobolev inequalities and contractivity properties of semigroups
,” in Dirichlet Forms (Varenna, 1992)
, Lecture Notes in Mathematics Vol. 1563 (Springer
, Berlin
, 1993
), pp. 54
–88
.4.
D.
Bakry
, “L’hypercontractivité et son utilisation en théorie des semigroupes
,” in Lectures on Probability Theory (Saint-Flour, 1992)
, Lecture Notes in Mathematics Vol. 1581 (Springer
, Berlin
, 1994
), pp. 1
–114
.5.
P.
Biane
, “Free hypercontractivity
,” Commun. Math. Phys.
184
, 457
–474
(1997
).6.
M.
Bożejko
and R.
Speicher
, “Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces
,” Math. Ann.
300
, 97
–120
(1994
).7.
M.
Bożejko
, “Ultracontractivity and strong Sobolev inequality for q-Ornstein–Uhlenbeck semigroup (−1 < q < 1)
,” Infinite Dimens. Anal. Quantum Probab. Relat. Top.
2
, 203
–220
(1999
).8.
I.
Królak
, “Contractivity properties of Ornstein-Uhlenbeck semigroup for general commutation relations
,” Math. Z.
250
, 915
–937
(2005
).9.
10.
H. H.
Lee
and É.
Ricard
, “Hypercontractivity on the q-Araki-Woods algebras
,” Commun. Math. Phys.
305
, 533
–553
(2011
).11.
F.
Hiai
, “q-deformed Araki-Woods algebras
,” in Operator Algebras and Mathematical Physics
(Theta
, Constanţa, Bucharest
, 2003
), pp. 169
–202
.12.
P.
Bikram
, R.
Kumar
, and K. K.
Mukherjee
, “Non injectivity of mixed q-deformed von Neumann algebras
,” (submitted).13.
P.
Bikram
and K.
Mukherjee
, “Generator masas in q-deformed Araki-Woods von Neumann algebras and factoriality
,” J. Funct. Anal.
273
, 1443
–1478
(2017
).14.
É.
Ricard
, “Factoriality of q-Gaussian von Neumann algebras
,” Commun. Math. Phys.
257
, 659
–665
(2005
).15.
S.
Avsec
, M.
Brannan
, and M.
Wasilewski
, “Complete metric approximation property for q-Araki-Woods algebras
,” J. Funct. Anal.
274
, 544
–572
(2018
).16.
S.
Avsec
, “Strong solidity of the q-Gaussian algebras for all −1 < q < 1
,” arXiv:1110.4918 [math.OA] (2012
).17.
A.
Nou
, “Non injectivity of the q-deformed von Neumann algebra
,” Math. Ann.
330
, 17
–38
(2004
).18.
C.
Houdayer
and É.
Ricard
, “Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors
,” Adv. Math.
228
, 764
–802
(2011
).19.
H.
Kosaki
, “Applications of the complex interpolation method to a von Neumann algebra: Noncommutative Lp-spaces
,” J. Funct. Anal.
56
, 29
–78
(1984
).20.
A.
Nou
, “Asymptotic matricial models and QWEP property for q-Araki–Woods algebras
,” J. Funct. Anal.
232
, 295
–327
(2006
).21.
M.
Junge
and Q.
Xu
, “Noncommutative maximal ergodic theorems
,” J. Am. Math. Soc.
20
, 385
–439
(2007
).22.
M.
Terp
, Lp-Spaces Associated with von Neumann Algebras
, Notes Mathematical Institute (University of Copenhagen
, 1981
), Vol. 3, p. 5
.23.
R.
Speicher
, “Generalized statistics of macroscopic fields
,” Lett. Math. Phys.
27
, 97
–104
(1993
).© 2021 Author(s). Published under an exclusive license by AIP Publishing.
2021
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