Commutative algebras of Toeplitz operators on the Reinhardt domains

TL;DR

This paper proves that Toeplitz operators with separately radial symbols on certain Reinhardt domains generate a commutative $C^*$-algebra, revealing geometric structures like foliations with Riemannian and Lagrangian properties.

Contribution

It generalizes the commutativity of Toeplitz operators from the unit disk to higher-dimensional Reinhardt domains and explores their geometric foliation structures.

Findings

The $C^*$-algebra generated by these Toeplitz operators is commutative.

The torus action induces a foliation with a transverse Riemannian structure.

The foliations are Lagrangian and have geometric properties like being equidistant and totally geodesic.

Abstract

Let $D$ be a bounded logarithmically convex complete Reinhardt domain in $\mathbb{C}^n$ centered at the origin. Generalizing a result for the one-dimensional case of the unit disk, we prove that the $C^*$-algebra generated by Toeplitz operators with bounded measurable separately radial symbols (i.e., symbols depending only on $|z_1|$, $|z_2|$, ..., $|z_n|$) is commutative. We show that the natural action of the $n$-dimensional torus $\mathbb{T}^n$ defines (on a certain open full measure subset of $D$) a foliation which carries a transverse Riemannian structure having distinguished geometric features. Its leaves are equidistant with respect to the Bergman metric, and the orthogonal complement to the tangent bundle of such leaves is integrable to a totally geodesic foliation. Furthermore, these two foliations are proved to be Lagrangian. We specify then the obtained results for the unit ball.