Duality, convexity and peak interpolation in the Drury–Arveson space

Abstract

We consider the closed algebra A d generated by the polynomial multipliers on the Drury-Arveson space. We identify A * d as a direct sum of the preduals of the full multiplier algebra and of a commutative von Neumann algebra, and establish analogues of many classical results concerning the dual space of the ball algebra. These developments are deeply intertwined with the problem of peak interpolation for multipliers, and we generalize a theorem of Bishop-Carleson-Rudin to this setting by means of Choquet type integral representations. As a byproduct we shed some light on the nature of the extreme points of the unit ball of A * d .2010 Mathematics Subject Classification. 47L30, 47A13, 46E22.

A multiplier algebra functional calculus

“…Results of this type were proved by J. Eschmeier, who established an H ∞ (B d )functional calculus for completely non-unitary tuples of commuting operators satisfying von Neumann's inequality over the unit ball [19] and by R. Clouâtre and K. Davidson, who studied the Drury-Arveson space H 2 d on B d and its multiplier algebra Mult(H 2 d ). They proved that completely non-unitary commuting row contractions admit a Mult(H 2 d )-functional calculus [13]. This current paper generalizes these earlier investigations and provides new arguments that avoid technical analyses of specific multiplier algebras.…”

Henkin Measures for the Drury–Arveson Space

“…has no spherical unitary summand), see [6,Theorem 4.1]. The following result is then a combination of Lemma 3.1 and Theorem 4.3 of [6]. This result shows that for the theory of commuting row contractions, A d -Henkin measures are a more suitable generalization of absolutely continuous measures on the unit circle than classical Henkin measures.…”

“…extends to a weak- * continuous functional on M d (see Subsection 2.1 for the definition of weak- * topology). Equivalently, whenever (f n ) is a sequence in A d such that ||f n || M d ≤ 1 for all n ∈ N and lim n→∞ f n (z) = 0 for all z ∈ B d , then Compelling evidence of the importance of A d -Henkin measures in multivariable operator theory can be found in [6], where Clouâtre and Davidson extend the Sz.-Nagy-Foias H ∞ -functional calculus to commuting row contractions. Recall that every contraction T on a Hilbert space can be written as T = T cnu ⊕ U , where U is a unitary operator and T cnu is completely nonunitary (i.e.…”

Multiplier algebras of complete Nevanlinna–Pick spaces: Dilations, boundary representations and hyperrigidity

“…That is, the operator algebra A(H) is the norm closure of the polynomials inside the multiplier algebra Mult(H [22,23,20,19]).…”