  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.
