Ideals in a multiplier algebra on the ball

TL;DR

This paper investigates the structure of ideals in the multiplier algebra on the Drury-Arveson space, exploring their closures, zero sets, and duality properties to extend classical results from the disc algebra.

Contribution

It provides new structural insights into the ideals of polynomial multipliers on the Drury-Arveson space, linking zero sets and duality in a novel way.

Findings

Characterization of ideals via zero sets

Relationship between an ideal and its weak-* closure

Structural parallels with classical disc algebra

Abstract

We study the ideals of the closure of the polynomial multipliers on the Drury-Arveson space. Structural results are obtained by investigating the relation between an ideal and its weak-$*$ closure, much in the spirit of the corresponding classical facts for the disc algebra. Zero sets for multipliers are also considered and are deeply intertwined with the structure of ideals. Our approach is primarily based on duality arguments.