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

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 $\ast$-representations of the ambient $C^{\ast }$-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 $C^{\ast }$-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.

40 pages; minor changes

References in corpus (3)

• The isomorphism problem for some universal operator algebras
• p-Summable Commutators in Dimension d
• Nevanlinna-Pick Kernels and Localization

Cited by in corpus (9)

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• Factorizations induced by complete Nevanlinna-Pick factors
• von Neumann's inequality for row contractive matrix tuples
• Non-commutative peaking phenomena and a local version of the hyperrigidity conjecture
• A Beurling-Lax-Halmos theorem for spaces with a complete Nevanlinna-Pick factor
• A simple proof of necessity in the McCullough-Quiggin theorem
• Operator inequalities I. Models and ergodicity
• Commutant lifting in several variables
• Finite-dimensionality in the non-commutative Choquet boundary: peaking phenomena and ${\mathrm{C}}^{\ast }$-liminality