Quantum exchangeable sequences of algebras Stephen Curran 46L5446L6560G09free probabilityquantum exchangeabilityquantum permutation groupquantum invariance We extend the notion of quantum exchangeability, introduced by K\"{o}stler and Speicher in [K. K\"{o}stler and R. Speicher, \emph{A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation}, http://www.arxiv.org/abs/0807.0677 (Preprint)], to sequences $(\rho_1,\rho_2,\dotsc)$ of homomorphisms from an algebra $C$ into a noncommutative probability space $(A,\varphi)$, and prove a free de Finetti theorem in this context: an infinite quantum exchangeable sequence $(\rho_1,\rho_2,\dotsc)$ is freely independent and identically distributed with respect to a conditional expectation. As in the classical case, the theorem fails for finite sequences. We give a characterization of finite quantum exchangeable sequences, which can be viewed as a noncommutative analogue of the classical urn sequences. We then give an approximation to how far a finite quantum exchangeable sequence is from being freely independent with amalgamation. Indiana University Mathematics Journal 2009 text pdf 10.1512/iumj.2009.58.3939 10.1512/iumj.2009.58.3939 en Indiana Univ. Math. J. 58 (2009) 1097 - 1126 state-of-the-art mathematics http://iumj.org/access/