Noncommutative peak interpolation revisited

TL;DR

This paper reviews and advances the theory of noncommutative peak interpolation, focusing on operator algebras on Hilbert spaces, providing simplified proofs of key theorems and introducing a new interpolation result.

Contribution

It offers simplified proofs of foundational theorems in noncommutative peak interpolation and introduces a new theorem, advancing the theoretical framework.

Findings

Simplified proofs of Hay and Read's theorems

Development of a new noncommutative peak interpolation theorem

Examples demonstrating applications of key theorems

Abstract

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 were spectacular, but therefore inaccessible to an uncommitted reader. We give short proofs of these results, using recent progress in noncommutative peak interpolation, and conversely give examples of the use of these theorems in peak interpolation. For example, we prove a useful new noncommutative peak interpolation theorem.