Structure and randomness in II_1 factors

Lundi 20 mars 16:00-17:00 Sorin Popa - University of California, Los Angeles

Résumé : II_1 factors are non-commutative versions of the function algebra L^\infty([0,1]),
the way matrix algebras M_{n\times n}(\mathbf C) are analogue to finite spaces. They arise as infinite tensor products and ultra products of matrix algebras, but also from groups \Gamma and their actions on probability spaces \Gamma \curvearrowright X. A key analysis tool to study II_1 factors in terms of their building data is \it deformation-rigidity theory. It fits within the fundamental dichotomy \it structure versus randomness, which appeared in many areas of mathematics in recent years. I will comment on this technique and present several classification results obtained this way, showing for instance that factors arising from Bernoulli actions of property (T) groups \Gamma \curvearrowright X ``remember’’ both the group and the action, and that free ergodic actions of the free groups \mathbf F_n remember the rank n.

Lieu : Bât. 425, petit amphi

Structure and randomness in II_1 factors  Version PDF