  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.
