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Remarks on factoriality and $ q$-deformations


Authors: Adam Skalski and Simeng Wang
Journal: Proc. Amer. Math. Soc. 146 (2018), 3813-3823
MSC (2010): Primary 46L36, 46L53, 81S05
DOI: https://doi.org/10.1090/proc/13715
Published electronically: June 1, 2018
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Abstract: We prove that the mixed $ q$-Gaussian algebra $ \Gamma _{Q}(H_{\mathbb{R}})$ associated to a real Hilbert space $ H_{\mathbb{R}}$ and a real symmetric matrix $ Q=(q_{ij})$ with $ \sup \vert q_{ij}\vert<1$, is a factor as soon as $ \dim H_{\mathbb{R}}\geq 2$. We also discuss the factoriality of $ q$-deformed Araki-Woods algebras, in particular showing that the $ q$-deformed Araki-Woods algebra $ \Gamma _{q}(H_{\mathbb{R}},U_{t})$ given by a real Hilbert space $ H_{\mathbb{R}}$ and a strongly continuous group $ U_{t}$ is a factor when $ \dim H_{\mathbb{R}}\geq 2$ and $ U_{t}$ admits an invariant eigenvector.


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Additional Information

Adam Skalski
Affiliation: Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00–656 Warszawa, Poland
Email: a.skalski@impan.pl

Simeng Wang
Affiliation: Laboratoire de Mathématiques, Université de Franche-Comté, 25030 Besançon Cedex, France – and – Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00–956 Warszawa, Poland
Address at time of publication: Universität des Saarlandes, FR 6.1-Mathematik, 66123 Saarbrücken, Germany
Email: wang@math.uni-sb.de

DOI: https://doi.org/10.1090/proc/13715
Received by editor(s): August 19, 2016
Received by editor(s) in revised form: February 15, 2017
Published electronically: June 1, 2018
Additional Notes: The authors were partially supported by the NCN (National Centre of Science) grant 2014/14/E/ST1/00525.
Communicated by: Adrian Ioana
Article copyright: © Copyright 2018 American Mathematical Society

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