A class of representations of $C^*$-algebra generated by $q_{ij}$-commuting isometries

Abstract

For a $C^*$-algebra generated by a finite family of isometries $s_j$, $j=1,\dots,d$, satisfying the $q_{ij}$-commutation relations \[ s_j^* s_j = I, \quad s_j^* s_k = q_{ij}s_ks_j^*, \qquad q_{ij} = \bar q_{ji}, |q_{ij}|<1, \ 1\le i \ne j \le d, \] we construct an infinite family of unitarily non-equivalent irreducible representations. These representations are deformations of a corresponding class of representations of the Cuntz algebra $\mathcal O_d$.

Для $C^*$-алгебри, породженої скінченною сім’єю ізометрій $s_j$, $j=1,\dots,d$, що задовольняє $q_{ij}$-комутаційним співвідношенням \[ s_j^* s_j = I, \quad s_j^* s_k = q_{ij}s_ks_j^*, \qquad q_{ij} = \bar q_{ji}, |q_{ij}|<1, \ 1\le i \ne j \le d, \] ми будуємо нескінченну сім'ю унітарно нееквівалентних незвідних представлень. Ці представлення є деформаціями відповідного класу представлень алгебри Кунца $\mathcal O_d$.

Key words: Cuntz-Toeplitz algebra, $q$-deformation, isometries.

Full Text

Article Information

Title	A class of representations of $C^*$-algebra generated by $q_{ij}$-commuting isometries
Source	Methods Funct. Anal. Topology, Vol. 28 (2022), no. 1, 89-94
MathSciNet	MR4459186
Milestones	Received 26/11/2021; Revised 05/12/2021
Copyright	The Author(s) 2022 (CC BY-SA)

Authors Information

Olha Ostrovska
National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”

Vasyl Ostrovskyi
Institute of Mathematics, NAS of Ukraine

Danylo Proskurin
Kyiv National Taras Shevchenko University

Yurii Samoilenko
Institute of Mathematics, NAS of Ukraine

Citation Example

Olha Ostrovska, Vasyl Ostrovskyi, Danylo Proskurin, and Yurii Samoilenko, A class of representations of $C^*$-algebra generated by $q_{ij}$-commuting isometries, Methods Funct. Anal. Topology 28 (2022), no. 1, 89-94.

BibTex

@article {MFAT1728,
    AUTHOR = {Olha Ostrovska and Vasyl Ostrovskyi and Danylo Proskurin and Yurii Samoilenko},
     TITLE = {A class of representations of $C^*$-algebra generated by $q_{ij}$-commuting isometries},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {28},
      YEAR = {2022},
    NUMBER = {1},
     PAGES = {89-94},
      ISSN = {1029-3531},
  MRNUMBER = {MR4459186},
       URL = {http://mfat.imath.kiev.ua/article/?id=1728},
}

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