Noncommutative Independence from Characters of the Infinite Symmetric Group $\mathbb{s}_\infty$

Details Noncommutative Independence from Characters of the Infinite Symmetric Group $\mathbb{s}_\infty$ Noncommutative Independence from Characters of the Infinite Symmetric Group $\mathbb{s}_\infty$

2010-05-31

arxiv.org/abs/1005.5726v1

Mathematics

Operator Algebras

47 pages

Scientific paper

We provide an operator algebraic proof of a classical theorem of Thoma which characterizes the extremal characters of the infinite symmetric group $\mathbb{S}_\infty$. Our methods are based on noncommutative conditional independence emerging from exchangeability and we reinterpret Thoma's theorem as a noncommutative de Finetti type result. Our approach is, in parts, inspired by Jones' subfactor theory and by Okounkov's spectral proof of Thoma's theorem, and we link them by inferring spectral properties from certain commuting squares.

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