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

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.