Multiplier algebras of complete Nevanlinna-Pick spaces: dilations, boundary representations and hyperrigidity

TL;DR

This paper investigates the structure of multiplier algebras of complete Nevanlinna-Pick spaces, providing a comprehensive analysis of their representations, dilations, boundary representations, and hyperrigidity, with extensions to subvarieties of the ball.

Contribution

It offers a complete description of representations, characterizes the Nevanlinna-Pick property via dilation phenomena, and identifies boundary representations and $C^*$-envelopes for these spaces.

Findings

Representations always dilate to $*$-representations with coextensions.

The Nevanlinna-Pick property is characterized by automatic coextensions.

Identified boundary representations and computed $C^*$-envelopes.

Abstract

We study reproducing kernel Hilbert spaces on the unit ball with the complete Nevanlinna-Pick property through the representation theory of their algebras of multipliers. We give a complete description of the representations in terms of the reproducing kernels. While representations always dilate to $*$-representations of the ambient $C^*$-algebra, we show that in our setting we automatically obtain coextensions. In fact, we show that in many cases, this phenomenon characterizes the complete Nevanlinna-Pick property. We also deduce operator theoretic dilation results which are in the spirit of work of Agler and several other authors. Moreover, we identify all boundary representations, compute the $C^*$-envelopes and determine hyperrigidity for certain analogues of the disc algebra. Finally, we extend these results to spaces of functions on homogeneous subvarieties of the ball.