Annals of Functional Analysis

Unitary representations of infinite wreath products

Robert P. Boyer and Yun S. Yoo

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Using C-algebraic techniques and especially AF-algebras, we present a complete classification of the continuous unitary representations for a class of infinite wreath product groups. These nonlocally compact groups are realized by a topological completion of the semidirect product of the countably infinite symmetric group acting on the countable direct product of a finite Abelian group.

Article information

Source
Ann. Funct. Anal., Volume 10, Number 1 (2019), 97-105.

Dates
Received: 19 September 2017
Accepted: 9 May 2018
First available in Project Euclid: 16 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.afa/1547629226

Digital Object Identifier
doi:10.1215/20088752-2018-0011

Mathematical Reviews number (MathSciNet)
MR3899959

Zentralblatt MATH identifier
07045488

Subjects
Primary: 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx]
Secondary: 20C32: Representations of infinite symmetric groups 20C99: None of the above, but in this section 43A40: Character groups and dual objects 46L05: General theory of $C^*$-algebras

Keywords
wreath product Littlewood–Richardson rule group algebra primitive ideal postliminary

Citation

Boyer, Robert P.; Yoo, Yun S. Unitary representations of infinite wreath products. Ann. Funct. Anal. 10 (2019), no. 1, 97--105. doi:10.1215/20088752-2018-0011. https://projecteuclid.org/euclid.afa/1547629226


Export citation

References

  • [1] D. Beltiţă and K.-H. Neeb, Schur-Weyl theory for $C^{*}$-algebras, Math. Nachr. 285 (2012), no. 10, 1170–1198.
  • [2] R. P. Boyer, “Representation theory of infinite-dimensional unitary groups” in Representation Theory of Groups and Algebras, Contemp. Math. 145, Amer. Math. Soc., Providence, 1993, 381–391.
  • [3] R. P. Boyer, Character theory of infinite wreath products, Int. J. Math. Math. Sci. 2005, no. 9, 1365–1379.
  • [4] F. Ingram, N. Jing, and E. Stitzinger, Wreath product symmetric functions, Int. J. Algebra 3 (2009), no. 1–4, 1–19.
  • [5] V. I. Kolomytsev and Y. S. Samoilenko, Irreducible representations of inductive limits of groups (in Russian), Ukraïn. Mat. Zh. 29, no. 4 (1977), 526–531; English translation in Ukrainian Math. J. 29 (1977), 402–405.
  • [6] K.-H. Neeb, “Unitary representations of unitary groups” in Developments and Retrospectives in Lie Theory, Dev. Math. 37, Springer, Cham, 2014, 197–243.
  • [7] N. Obata, Some remarks on induced representations of infinite discrete groups, Math. Ann. 284 (1989), no. 1, 91–102.
  • [8] G. I. Olshanski, “Unitary representations of the infinite symmetric group: A semigroup approach” in Representations of Lie Groups and Lie Algebras (Budapest, 1971), Akad. Kiadó, Budapest, 1985, 181–197.
  • [9] Ş. Strătilă and D. Voiculescu, Representations of AF-Algebras and of the Group $U(\infty)$, Lecture Notes in Math. 486, Springer, Berlin, 1975.
  • [10] A. M. Vershik and S. V. Kerov, Locally semisimple algebras: Combinatorial theory and the $K_{0}$-functor (in Russian), Itogi Nauki Tekh., Sovrem. Probl. Mat. 26 (1985), 3–56; English translation in J. Soviet Math. 38 (1987), no. 2, 1701–1733.
  • [11] A. V. Zelevinsky, Representations of Finite Classical Groups: A Hopf Algebra Approach, Lecture Notes in Math. 869, Springer, Berlin, 1981.