Remarks on factoriality and $q$-deformations
HTML articles powered by AMS MathViewer
- by Adam Skalski and Simeng Wang PDF
- Proc. Amer. Math. Soc. 146 (2018), 3813-3823 Request permission
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 $\sup |q_{ij}|<1$, 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.References
- 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
- Panchugopal Bikram and Kunal Mukherjee, Generator masas in $q$-deformed Araki-Woods von Neumann algebras and factoriality, J. Funct. Anal. 273 (2017), no. 4, 1443–1478. MR 3661405, DOI 10.1016/j.jfa.2017.03.005
- 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
- Fumio Hiai, $q$-deformed Araki-Woods algebras, Operator algebras and mathematical physics (Constanţa, 2001) Theta, Bucharest, 2003, pp. 169–202. MR 2018229
- 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, Factoriality of von Neumann algebras connected with general commutation relations—finite dimensional case, Quantum probability, Banach Center Publ., vol. 73, Polish Acad. Sci. Inst. Math., Warsaw, 2006, pp. 277–284. MR 2423133, DOI 10.4064/bc73-0-20
- 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, Asymptotic matricial models and QWEP property for $q$-Araki–Woods algebras, J. Funct. Anal. 232 (2006), no. 2, 295–327. MR 2200739, DOI 10.1016/j.jfa.2005.05.001
- Brent Nelson and Qiang Zeng, An application of free transport to mixed $q$-Gaussian algebras, Proc. Amer. Math. Soc. 144 (2016), no. 10, 4357–4366. MR 3531185, DOI 10.1090/proc/13068
- 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, 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
- Mateusz Wasilewski, $q$-Araki-Woods algebras: extension of second quantisation and Haagerup approximation property, Proc. Amer. Math. Soc. 145 (2017), no. 12, 5287–5298. MR 3717957, DOI 10.1090/proc/13681
Additional Information
- Adam Skalski
- Affiliation: Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00–656 Warszawa, Poland
- MR Author ID: 705797
- ORCID: 0000-0003-1661-8369
- 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
- MR Author ID: 1186269
- Email: wang@math.uni-sb.de
- 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
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 3813-3823
- MSC (2010): Primary 46L36, 46L53, 81S05
- DOI: https://doi.org/10.1090/proc/13715
- MathSciNet review: 3825836