Seminario de geometria diferencial y sistemas dinamicos, CIMAT

Fecha y lugar: Lunes 9 mar 2020, 4:45pm, Salon Diego Bricio

Speaker: Raul Quiroga, CIMAT

Título: Toeplitz operators and Kähler geometry.

Resúmen: The complex n-dimensional unit ball B^n carries
rich structures from both the analytic and geometric viewpoints.
Analytically, B^n is the domain of a continuous family of
reproducing kernel Hilbert spaces, the so-called weighted Bergman
spaces. Toeplitz operators are naturally defined bounded operators
acting on these Bergman spaces. Geometrically, B^n is a
Hermitian symmetric space with holomorphic isometry group given by
SU(n,1), and so it carries a nice Kähler geometry. For n=1, we have
the unit disk D and it is known that every one-parameter
subgroup of SU(1,1) yields a commutative C*-algebra generated by
(invariant) Toeplitz operators. We will discuss how Analysis and
Geometry can be used to prove that, under some non-triviality
conditions, all commutative C*-algebras generated by Toeplitz operators
are given by some one-parameter subgroup of SU(1,1) for the case of
the unit disk. We will also discuss some advances in the solution of the
corresponding problem for the general case of the unit ball B^n.