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 ⁎-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.