An application of free transport to mixed $q$-Gaussian algebras
HTML articles powered by AMS MathViewer
- by Brent Nelson and Qiang Zeng PDF
- Proc. Amer. Math. Soc. 144 (2016), 4357-4366 Request permission
Abstract:
We consider the mixed $q$-Gaussian algebras introduced by Speicher which are generated by the variables $X_i=l_i+l_i^*,i=1,\ldots ,N$, where $l_i^* l_j-q_{ij}l_j l_i^*=\delta _{i,j}$ and $-1<q_{ij}=q_{ji}<1$. Using the free monotone transport theorem of Guionnet and Shlyakhtenko, we show that the mixed $q$-Gaussian von Neumann algebras are isomorphic to the free group von Neumann algebra $L(\mathbb {F}_N)$, provided that $\max _{i,j}|q_{ij}|$ is small enough. The proof relies on some estimates which are generalizations of Dabrowski’s results for the special case $q_{ij}\equiv q$.References
- S. Avsec, Strong Solidity of the q-Gaussian Algebras for all $-1 < q < 1$, ArXiv e-prints (2011-10), available at 1110.4918.
- Marek Bożejko, Burkhard Kümmerer, and Roland Speicher, $q$-Gaussian processes: non-commutative and classical aspects, Comm. Math. Phys. 185 (1997), no. 1, 129–154. MR 1463036, DOI 10.1007/s002200050084
- Marek Bożejko, Completely positive maps on Coxeter groups and the ultracontractivity of the $q$-Ornstein-Uhlenbeck semigroup, Quantum probability (Gdańsk, 1997) Banach Center Publ., vol. 43, Polish Acad. Sci. Inst. Math., Warsaw, 1998, pp. 87–93. MR 1649711
- Marek Bożejko and Roland Speicher, An example of a generalized Brownian motion, Comm. Math. Phys. 137 (1991), no. 3, 519–531. MR 1105428
- Marek Bożejko and Roland Speicher, Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces, Math. Ann. 300 (1994), no. 1, 97–120. MR 1289833, DOI 10.1007/BF01450478
- Yoann Dabrowski, A free stochastic partial differential equation, Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), no. 4, 1404–1455. MR 3270000, DOI 10.1214/13-AIHP548
- A. Guionnet and D. Shlyakhtenko, Free monotone transport, Invent. Math. 197 (2014), no. 3, 613–661. MR 3251831, DOI 10.1007/s00222-013-0493-9
- Fumio Hiai, $q$-deformed Araki-Woods algebras, Operator algebras and mathematical physics (Constanţa, 2001) Theta, Bucharest, 2003, pp. 169–202. MR 2018229
- M. Junge and Q. Zeng, Mixed $q$-Gaussian algebras, ArXiv e-prints (2015-05), available at 1505.07852.
- Matthew Kennedy and Alexandru Nica, Exactness of the Fock space representation of the $q$-commutation relations, Comm. Math. Phys. 308 (2011), no. 1, 115–132. MR 2842972, DOI 10.1007/s00220-011-1323-9
- Ilona Krȯlak, Wick product for commutation relations connected with Yang-Baxter operators and new constructions of factors, Comm. Math. Phys. 210 (2000), no. 3, 685–701. MR 1777345, DOI 10.1007/s002200050796
- Ilona Królak, Contractivity properties of Ornstein-Uhlenbeck semigroup for general commutation relations, Math. Z. 250 (2005), no. 4, 915–937. MR 2180382, DOI 10.1007/s00209-005-0801-1
- Françoise Lust-Piquard, Riesz transforms on deformed Fock spaces, Comm. Math. Phys. 205 (1999), no. 3, 519–549. MR 1711277, DOI 10.1007/s002200050688
- Brent Nelson, Free monotone transport without a trace, Comm. Math. Phys. 334 (2015), no. 3, 1245–1298. MR 3312436, DOI 10.1007/s00220-014-2148-0
- Alexandre Nou, Non injectivity of the $q$-deformed von Neumann algebra, Math. Ann. 330 (2004), no. 1, 17–38. MR 2091676, DOI 10.1007/s00208-004-0523-4
- Narutaka Ozawa and Sorin Popa, On a class of $\textrm {II}_1$ factors with at most one Cartan subalgebra, Ann. of Math. (2) 172 (2010), no. 1, 713–749. MR 2680430, DOI 10.4007/annals.2010.172.713
- Éric Ricard, Factoriality of $q$-Gaussian von Neumann algebras, Comm. Math. Phys. 257 (2005), no. 3, 659–665. MR 2164947, DOI 10.1007/s00220-004-1266-5
- Dimitri Shlyakhtenko, Some estimates for non-microstates free entropy dimension with applications to $q$-semicircular families, Int. Math. Res. Not. 51 (2004), 2757–2772. MR 2130608, DOI 10.1155/S1073792804140476
- Dimitri Shlyakhtenko, Free quasi-free states, Pacific J. Math. 177 (1997), no. 2, 329–368. MR 1444786, DOI 10.2140/pjm.1997.177.329
- Piotr Śniady, Factoriality of Bożejko-Speicher von Neumann algebras, Comm. Math. Phys. 246 (2004), no. 3, 561–567. MR 2053944, DOI 10.1007/s00220-003-1031-1
- Roland Speicher, Generalized statistics of macroscopic fields, Lett. Math. Phys. 27 (1993), no. 2, 97–104. MR 1213608, DOI 10.1007/BF00750677
- D. V. Voiculescu, K. J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. MR 1217253, DOI 10.1090/crmm/001
- Dan Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. V. Noncommutative Hilbert transforms, Invent. Math. 132 (1998), no. 1, 189–227. MR 1618636, DOI 10.1007/s002220050222
Additional Information
- Brent Nelson
- Affiliation: Department of Mathematics, University of California, Berkeley, California 94709
- MR Author ID: 926329
- Email: brent@math.berkeley.edu
- Qiang Zeng
- Affiliation: Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts 02138
- Address at time of publication: Mathematics Department, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208
- Email: qzeng.math@gmail.com
- Received by editor(s): July 23, 2015
- Received by editor(s) in revised form: December 12, 2015
- Published electronically: April 13, 2016
- Additional Notes: The research of the first author was supported by the NSF awards DMS-1161411 and DMS-1502822.
- Communicated by: Adrian Ioana
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4357-4366
- MSC (2010): Primary 46L54, 81S05
- DOI: https://doi.org/10.1090/proc/13068
- MathSciNet review: 3531185