Gelfand transforms and boundary representations of complete Nevanlinna--Pick quotients

TL;DR

This paper investigates the boundary representations of quotients of multiplier algebras in complete Nevanlinna--Pick spaces, linking non-commutative boundaries with Gelfand transforms and essential normality, extending Arveson-Douglas conjecture results.

Contribution

It establishes a connection between hyperrigidity, essential normality, and non-commutative Choquet boundaries in these quotient algebras, providing new insights into their structure.

Findings

Hyperrigidity is detectable via essential normality.

Non-commutative Choquet boundaries relate to Gelfand transform isometry.

Analytic and topological conditions clarify boundary structure.

Abstract

The main objects under study are quotients of multiplier algebras of certain complete Nevanlinna--Pick spaces, examples of which include the Drury--Arveson space on the ball and the Dirichlet space on the disc. We are particularly interested in the non-commutative Choquet boundaries for these quotients. Arveson's notion of hyperrigidity is shown to be detectable through the essential normality of some natural multiplication operators, thus extending previously known results on the Arveson--Douglas conjecture. We also highlight how the non-commutative Choquet boundaries of these quotients are intertwined with their Gelfand transforms being completely isometric. Finally, we isolate analytic and topological conditions on the so-called supports of the underlying ideals that clarify the nature of the non-commutative Choquet boundaries.