## Annals of Functional Analysis

### Unitary representations of infinite wreath products

#### 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
Accepted: 9 May 2018
First available in Project Euclid: 16 January 2019

https://projecteuclid.org/euclid.afa/1547629226

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

Mathematical Reviews number (MathSciNet)
MR3899959

Zentralblatt MATH identifier
07045488

#### 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

#### 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.