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

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.