A. V. Dudko

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Articles: 1

A description of characters on the infinite wreath product

A. V. Dudko, N. I. Nessonov

↓ Abstract   |   Article (.pdf)

MFAT 13 (2007), no. 4, 301-317


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 $\mathfrak{S}_\infty$.

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