## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Friday, February 5, 2016 — 3:30 PM EST

**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**

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.