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The main objects under study are quotients of multiplier algebras of certaincomplete Nevanlinna--Pick spaces, examples of which include the Drury--Arvesonspace on the ball and the Dirichlet space on the disc. We are particularlyinterested in the non-commutative Choquet boundaries for these quotients.Arveson's notion of hyperrigidity is shown to be detectable through theessential normality of some natural multiplication operators, thus extendingpreviously known results on the Arveson--Douglas conjecture. We also highlighthow the non-commutative Choquet boundaries of these quotients are intertwinedwith their Gelfand transforms being completely isometric. Finally, we isolateanalytic and topological conditions on the so-called supports of the underlyingideals that clarify the nature of the non-commutative Choquet boundaries.

Gelfand transforms and boundary representations of complete Nevanlinna--Pick quotients