Cumulants in noncommutative probability theory IV. Noncrossing cumulants: De Finetti's theorem and Lp-inequalities

De Finetti's theorem states that any exchangeable sequence of classical random variables is conditionally i.i.d. with respect to some σ-algebra. In this paper we prove a “free” noncommutative analog of this theorem, namely we show that any noncrossing exchangeability system with a faithful state which satisfies a so called weak singleton condition can be embedded into an free product with amalgamation over a certain subalgebra such that the interchangeable algebras remain interchangeable with respect to the operator-valued expectation. Vanishing of crossing cumulants can be verified by checking a certain weak freeness condition and the weak singleton condition is satisfied e.g. when the state is tracial. The proof follows the classical proof of De Finetti's theorem, the main technical tool being a noncommutative Lp-inequality for i.i.d. sums of centered noncommutative random variables in noncrossing exchangeability systems.