Gelfand transforms and boundary representations of complete Nevanlinna–Pick quotients

Abstract

The main objects under study are quotients of multiplier algebras of certain complete Nevanlinna–Pick spaces, examples of which include the Drury–Arveson space on the ball and the Dirichlet space on the disc. We are particularly interested in the non-commutative Choquet boundaries for these quotients. Arveson’s notion of hyperrigidity is shown to be detectable through the essential normality of some natural multiplication operators, thus extending previously known results on the Arveson–Douglas conjecture. We also highlight how the non-commutative Choquet boundaries of these quotients are intertwined with their Gelfand transforms being completely isometric. Finally, we isolate analytic and topological conditions on the so-called supports of the underlying ideals that clarify the nature of the non-commutative Choquet boundaries.

Minimal boundaries for operator algebras

“…Examples of spaces to which this result applies include the Hardy space on the unit disc, the Dirichlet space on the unit disc, as well as the Drury-Arveson space on the unit ball. These spaces and their multiplier algebras have generated much research interest in recent years; see for instance [21], [22], [24], [27] and references therein. We also show in Theorem 4.6 that, under some regularity conditions, the Bishop property is preserved by tensor products, thus allowing us to produce more examples (Corollary 4.7).…”

“…The resulting class of regular, unitarily invariant, complete Pick spaces is highly structured and contains many frequently studied examples arising naturally in function theory and operator theory, such as the Hardy space on the disc, the classical Dirichlet space, as well as the Drury-Arveson space. The reader may consult [22], [24], [27] for further detail.…”