Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T11:58:54.750Z Has data issue: false hasContentIssue false
  • English
  • Français

On the von Neumann algebras associated to Yang–Baxter operators

Part of: Selfadjoint operator algebras Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  28 August 2020

Panchugopal Bikram ,
Rahul Kumar ,
Rajeeb Mohanta ,
Kunal Mukherjee  and
Diptesh Saha
Panchugopal Bikram
Affiliation:
School of Mathematical Sciences, National Institute of Science Education and Research, HBNI, Bhubaneswar, Odisha752050, India Department of Mathematics, IIT Madras, Chennai600036, India ([email protected], [email protected], [email protected], [email protected], [email protected])
Rahul Kumar
Affiliation:
School of Mathematical Sciences, National Institute of Science Education and Research, HBNI, Bhubaneswar, Odisha752050, India Department of Mathematics, IIT Madras, Chennai600036, India ([email protected], [email protected], [email protected], [email protected], [email protected])
Rajeeb Mohanta
Affiliation:
School of Mathematical Sciences, National Institute of Science Education and Research, HBNI, Bhubaneswar, Odisha752050, India Department of Mathematics, IIT Madras, Chennai600036, India ([email protected], [email protected], [email protected], [email protected], [email protected])
Kunal Mukherjee
Affiliation:
School of Mathematical Sciences, National Institute of Science Education and Research, HBNI, Bhubaneswar, Odisha752050, India Department of Mathematics, IIT Madras, Chennai600036, India ([email protected], [email protected], [email protected], [email protected], [email protected])
Diptesh Saha
Affiliation:
School of Mathematical Sciences, National Institute of Science Education and Research, HBNI, Bhubaneswar, Odisha752050, India Department of Mathematics, IIT Madras, Chennai600036, India ([email protected], [email protected], [email protected], [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Bożejko and Speicher associated a finite von Neumann algebra MT to a self-adjoint operator T on a complex Hilbert space of the form $\mathcal {H}\otimes \mathcal {H}$ which satisfies the Yang–Baxter relation and $ \left\| T \right\| < 1$. We show that if dim$(\mathcal {H})$ ⩾ 2, then MT is a factor when T admits an eigenvector of some special form.

Keywords

Yang–Baxter operatorvon Neumann algebras46L54

MSC classification

Primary: 46L10: General theory of von Neumann algebras 46L54: Free probability and free operator algebras
Secondary: 46L40: Automorphisms 46L53: Noncommutative probability and statistics 46L36: Classification of factors 46C99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 4 , August 2021 , pp. 1331 - 1354
Check for updates
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Avesec, S.. Strong solidity of the q-Gaussian algebras for all − 1 < q < 1, preprint, arXiv:1110.4918, (2011).Google Scholar
Bikram, P. and Mukherjee, K.. Generator masas in q-deformed Araki–Woods von Neumann algebras and factoriality. J. Funct. Anal., 273 (2017), 1443–1478.10.1016/j.jfa.2017.03.005CrossRefGoogle Scholar
Bikram, P., Kumar, R. and Mukherjee, K.. Mixed q-deformed Araki-Woods von Neumann algebras, preprint, (2020).Google Scholar
Bourbaki, N.. Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Hermann, Paris, 1968, 288 pp. (loose errata).Google Scholar
Bożejko, M. and Speicher, R.. Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces. Math. Ann., 300 (1994), 97–120.CrossRefGoogle Scholar
Bożejko, M., Kümmerer, B. and Speicher, R.. q -Gaussian processes: non-commutative and classical aspects. Comm. Math. Phys., 185 (1997), 129–154.Google Scholar
Cameron, J., Fang, J. and Mukherjee, K.. Mixing subalgebras of finite von Neumann algebras, New York. J. Math., 19 (2013), 343–366).Google Scholar
Junge, M. and Zeng, Q.. Mixed q-Gaussian algebras, unpublished, (2015).Google Scholar
Królak, I.. Wick product for commutation relations connected with Yang–Baxter operators and new constructions of factors. Comm. Math. Phys., 210 (2000), 685–701.Google Scholar
Królak, I.. Factoriality of von Neumann algebras connected with general commutation relations-finite dimensional case. Quantum Probability, Banach Center Publ., 73 (2006), 277–284.CrossRefGoogle Scholar
Mukherjee, K.. Singular masas and measure-multiplicity invariant. Houston J. Math., 39 (2013), 561–598.Google Scholar
Mukherjee, K.. Masas and bimodule decompositions of II1 factors. Quart. J. Math., 62 (2009), 451–486.CrossRefGoogle Scholar
Nelson, B. and Zeng, Q.. An application of free transport to mixed q-Gaussian algebras. Proc. Amer. Math. Soc., 144 (2016), 4357–4366.CrossRefGoogle Scholar
Nou, A.. Non injectivity of the q-deformed von Neumann algebra. Math. Ann., 330 (2004), 17–38.CrossRefGoogle Scholar
Ozawa, N.. Solid von Neumann algebras. Acta Math. 192 (2004), 111–117.CrossRefGoogle Scholar
Ricard, É.. Factoriality of q-Gaussian von NeumannAlgebras. Comm. Math. Phys. 257 (2005), 659–665.CrossRefGoogle Scholar
Voiculescu, D. V., Dykema, K. J. and Nica, A.. Free Random Variables, CRM Monograph Series, vol. 1, (Providence, RI: American Mathematical Society, 1992).CrossRefGoogle Scholar
Wang, S.. Singularity of the generator subalgebra in mixe d q-Gaussian algebras. Colloq. Math. 158 (2019), 39–43.CrossRefGoogle Scholar