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An application of free transport to mixed $ q$-Gaussian algebras


Authors: Brent Nelson and Qiang Zeng
Journal: Proc. Amer. Math. Soc. 144 (2016), 4357-4366
MSC (2010): Primary 46L54, 81S05
DOI: https://doi.org/10.1090/proc/13068
Published electronically: April 13, 2016
MathSciNet review: 3531185
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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}\vert q_{ij}\vert$ is small enough. The proof relies on some estimates which are generalizations of Dabrowski's results for the special case $ q_{ij}\equiv q$.


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Additional Information

Brent Nelson
Affiliation: Department of Mathematics, University of California, Berkeley, California 94709
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

DOI: https://doi.org/10.1090/proc/13068
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
Article copyright: © Copyright 2016 American Mathematical Society