Commutative $C^*$-algebras of Toeplitz operators on complex projective spaces

TL;DR

This paper demonstrates the existence of commutative $C^*$-algebras of Toeplitz operators on weighted Bergman spaces over complex projective spaces, linking algebraic structures to geometric properties.

Contribution

It introduces a new class of commutative $C^*$-algebras of Toeplitz operators defined by symbols depending on the radial part, connected to the geometry of complex projective spaces.

Findings

Existence of commutative $C^*$-algebras on weighted Bergman spaces

Algebras are generated by symbols depending only on radial coordinates

Associated with pairs of Lagrangian foliations with geometric significance

Abstract

We prove the existence of commutative $C^*$-algebras of Toeplitz operators on every weighted Bergman space over the complex projective space $\mathbb{P}^n(\mathbb{C})$. The symbols that define our algebras are those that depend only on the radial part of the homogeneous coordinates. The algebras presented have an associated pair of Lagrangian foliations with distinguished geometric properties and are closely related to the geometry of $\mathbb{P}^n(\mathbb{C})$.