**
Serban
Belinschi,
University
of
Toulouse,
France**

“A Julia-Caratheodory Theorem for noncommutative functions (and some applications)”

The Julia-Caratheodory Theorem describes the behaviour of the dervative of an analytic self-map of a ”good” domain near certain points of the boundary of the domain. For the upper half-plane, this theorem essentially states that boundedness of the quotient of the imaginary part of the function by the imaginary part of the variable along some sequence converging to a real number implies the existence and finiteness of the nontangential limit of the derivative of the function at the same point. This theorem has been generalized by numerous authors (to maps on polydisks, maps with values in algebras of operators, maps on bounded symmetric domains etc), and seems to be of quite some interest to this day. In this talk we shall present a generalization of the Julia-Caratheodory Theorem to noncommutative self-maps of the non- commutative upper half-plane of a von Neumann algebra. The proof makes use repeatedly and in essential ways of properties specific to noncommutative functions. Time permitting, we shall indicate some possible applications to noncommutative probability.

MC 2017 **Please note Room**