SAMS Subject Classification Number: 24

The investigation of distributional symmetries was initiated by de Finetti’s celebrated theorem, which shows that any finite joint distribution of sequences of two-point valued exchangeable random variables is obtained by randomization of the binomial distribution. This result has since found several generalizations both in classical and noncommutative settings. Also motivated by the key role played in physics by the CAR algebra, we carry out a careful study of the (minimal) infinite graded tensor product of a given \(C^*\)-algebra with itself, which is acted upon in a natural way by the group of finite permutations. Invariant states for this action turn out to be automatically even and extreme invariant states are characterized as infinite products of a single even state on the \(C^*\)-algebra. As a consequence, the extreme symmetric states of the (minimal) graded tensor product are sufficiently many to separate its points, allowing us to prove weak ergodicity of the permutation action. Finally, a version of de Finetti’s theorem for graded processes is established, for in this case invariant states correspond to exchangeable quantum stochastic processes.
The talk is based on joint work with V. Crismale and S. Rossi.