Quantum symmetric states on free product $C^*$-algebras

Abstract:

We introduce symmetric states and quantum symmetric states on universal unital free product $C^*$-algebras of the form $\mathfrak {A}=\operatorname * {\ast }_1^\infty A$ for an arbitrary unital $C^*$-algebra $A$ as a generalization of the notions of exchangeable and quantum exchangeable random variables. We prove the existence of conditional expectations onto tail algebras in various settings and we define a natural $C^*$-subalgebra of the tail algebra, called the tail $C^*$-algebra. Extending and building on the proof of the noncommutative de Finetti theorem of Köstler and Speicher, we prove a de Finetti type theorem that characterizes quantum symmetric states in terms of amalgamated free products over the tail $C^*$-algebra, and we provide a convenient description of the set of all quantum symmetric states on $\mathfrak {A}$ in terms of $C^*$-algebras generated by homomorphic images of $A$ and the tail $C^*$-algebra. This description allows a characterization of the extreme quantum symmetric states. Similar results are proved for the subset of tracial quantum symmetric states, though in terms of von Neumann algebras and normal conditional expectations. The central quantum symmetric states are those for which the tail algebra is in the center of the von Neumann algebra, and we show that the central quantum symmetric states form a Choquet simplex whose extreme points are the free product states, while the tracial central quantum symmetric states form a Choquet simplex whose extreme points are the free product traces.