Duality, convexity and peak interpolation in the Drury-Arveson space

TL;DR

This paper explores the dual space structure of the algebra of polynomial multipliers on the Drury-Arveson space, extending classical results and addressing peak interpolation problems with new integral representation techniques.

Contribution

It identifies the dual of the algebra as a direct sum of preduals and generalizes classical peak interpolation theorems to the Drury-Arveson setting.

Findings

Dual space of the algebra is a direct sum of preduals.

Generalization of Bishop-Carleson-Rudin theorem for multipliers.

Insights into extreme points of the dual unit ball.

Abstract

We consider the closed algebra $\mathcal{A}_d$ generated by the polynomial multipliers on the Drury-Arveson space. We identify $\mathcal{A}_d^*$ as a direct sum of the preduals of the full multiplier algebra and of a commutative von Neumann algebra, and establish analogues of many classical results concerning the dual space of the ball algebra. These developments are deeply intertwined with the problem of peak interpolation for multipliers, and we generalize a theorem of Bishop-Carleson-Rudin to this setting by means of Choquet type integral representations. As a byproduct we shed some light on the nature of the extreme points of the unit ball of $\mathcal{A}^*_d$.