Classifying Azema martingales: from probability to algebra and back.

Azema martingales arise naturally in the study of the chaotic representation property; they also provide classical interpretations of quantum stochastic calculus. The talk will not insist on these aspects, but only define these processes and address the problem of their classification. This raises algebraic questions concerning tensors. Everyone knows that matrices can be diagonalized in some common orthonormal basis if and only if they are symmetric and commute with each other; we shall see an analogous statement for tensors with more
than two indices. This, and other theorems in the same vein, make it possible to associate to any multidimensional Azema martingale an orthogonal decomposition of the state space into one- and two-dimensional subspaces; the behaviour of the process becomes simpler when split into its components in these sub-spaces.