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# A description of characters on the infinite wreath product

### Abstract

Let $\mathfrak{S}_\infty$ be the infinity permutation group and $\Gamma$ an arbitrary group. Then $\mathfrak{S}_\infty$ admits a natural action on $\Gamma^\infty$ by automorphisms, so one can form a semidirect product $\Gamma^\infty \times \mathfrak{S}_\infty$, known as the wreath product $\Gamma\wr\mathfrak{S}_\infty$ of $\Gamma$ by $\mathfrak{S}_{\infty}$. We obtain a full description of unitary $I\!I_1-$factor-representations of $\Gamma\wr\mathfrak{S}_\infty$ in terms of finite characters of $\Gamma$. Our approach is based on extending Okounkov's classification method for admissible representations of $\mathfrak{S}_\infty\times\mathfrak{S}_\infty$. Also, we discuss certain examples of representations of type $I\!I\!I$, where the modular operator of Tomita-Takesaki expresses naturally by the asymptotic operators, which are important in the theory of characters of $\mathfrak{S}_\infty$.

### Article Information

 Title A description of characters on the infinite wreath product Source Methods Funct. Anal. Topology, Vol. 13 (2007), no. 4, 301-317 MathSciNet MR2374832 Copyright The Author(s) 2007 (CC BY-SA)

### Authors Information

A. V. Dudko
Kharkiv National University, Kharkiv, Ukraine

N. I. Nessonov
Department of Mathematics, Institute For Low Temperature Physics and Engineering, 47 Lenin Avenue, Kharkiv, Ukraine

### Citation Example

A. V. Dudko and N. I. Nessonov, A description of characters on the infinite wreath product, Methods Funct. Anal. Topology 13 (2007), no. 4, 301-317.

### BibTex

@article {MFAT391,
AUTHOR = {Dudko, A. V. and Nessonov, N. I.},
TITLE = {A description of characters on the infinite wreath product},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {13},
YEAR = {2007},
NUMBER = {4},
PAGES = {301-317},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=391},
}