# Noncommutative Independence from Characters of the Infinite Symmetric Group $mathbb{s} infty$ - Mathematics > Operator Algebras

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Abstract: We provide an operator algebraic proof of a classical theorem of Thoma whichcharacterizes the extremal characters of the infinite symmetric group$\mathbb{S} \infty$. Our methods are based on noncommutative conditionalindependence emerging from exchangeability and we reinterpret Thoma-s theoremas a noncommutative de Finetti type result. Our approach is, in parts, inspiredby Jones- subfactor theory and by Okounkov-s spectral proof of Thoma-s theorem,and we link them by inferring spectral properties from certain commutingsquares.

Author: ** Rolf Gohm, Claus Köstler**

Source: https://arxiv.org/