  The extended de Finetti theorem characterizes exchangeable infinite random
sequences as conditionally i.i.d. and shows that the apparently weaker
distributional symmetry of spreadability is equivalent to exchangeability. Our
main result is a noncommutative version of this theorem.
  In contrast to the classical result of Ryll-Nadzewski, exchangeability turns
out to be stronger than spreadability for infinite noncommutative random
sequences. Out of our investigations emerges noncommutative conditional
independence in terms of a von Neumann algebraic structure closely related to
Popa's notion of commuting squares and K\"ummerer's generalized Bernoulli
shifts. Our main result is applicable to classical probability, quantum
probability, in particular free probability, braid group representations and
Jones subfactors.
