Closed projections and peak interpolation for operator algebras

非交换的 Bishop 峰值插值集定理

A noncommutative Bishop peak interpolation-set theorem

本文证明了Bishop峰值插值集定理的非交换版本，扩展了该经典插值定理在非交换框架下的适用性。

Noncommutative peak interpolation revisited

Peak interpolation is concerned with a foundational kind of mathematical task: building functions in a fixed algebra $A$ which have prescribed values or behaviour on a fixed closed subset (or on several disjoint subsets). In this paper we do the same but now $A$ is an algebra of operators on a Hilbert space. We briefly survey this {\em noncommutative peak interpolation}, which we have studied with coauthors in a long series of papers, and whose basic theory now appears to be approaching its culmination. This program developed from, and is based partly on, theorems of Hay and Read whose proofs …

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

We consider the closed algebra $\mathcal{A}_d$ generated by the polynomial multipliers on the Drury-Arveson space. We identify $\mathcal{A}_d^*$ 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 o…

Open projections in operator algebras II: Compact projections

We generalize some aspects of the theory of compact projections relative to a C*-algebra, to the setting of more general algebras. Our main result is that compact projections are the decreasing limits of `peak projections', and in the separable case compact projections are just the peak projections. We also establish new forms of the noncommutative Urysohn lemma relative to an operator algebra, and we show that a projection is compact iff the associated face in the state space of the algebra is weak* closed.

Interpolation and duality in algebras of multipliers on the ball

We study the multiplier algebras $A(\mathcal{H})$ obtained as the closure of the polynomials on certain reproducing kernel Hilbert spaces $\mathcal{H}$ on the ball $\mathbb{B}_d$ of $\mathbb{C}^d$. Our results apply, in particular, to the Drury-Arveson space, the Dirichlet space and the Hardy space on the ball. We first obtain a complete description of the dual and second dual spaces of $A(\mathcal H)$ in terms of the complementary bands of Henkin and totally singular measures for $\operatorname{Mult}(\mathcal{H})$. This is applied to obtain several definitive results in interpolation. In part…

算子代数中的零投影与非交换函数论

Null projections and noncommutative function theory in operator algebras

本文研究算子代数中关于子代数为零的投影，将其与峰值投影和经典零集类比，通过推广Varopoulos定理刻画零投影条件，并应用于峰值插值及非交换函数论多个主题。