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On the von Neumann algebras associated to Yang–Baxter operators

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

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])
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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
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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