Minimal boundaries for operator algebras

TL;DR

This paper explores the concept of boundaries in unital operator algebras, extending classical ideas of the Choquet boundary to the non-commutative setting, and introduces the Bishop property as a key criterion for minimality.

Contribution

It establishes the equivalence between minimality of the non-commutative boundary and the Bishop property, providing new insights and proofs in the theory of operator algebras.

Findings

Minimality of non-commutative boundaries is equivalent to the Bishop property.

Not all operator algebras possess the Bishop property; examples are provided.

A new proof characterizes $C^*$-algebras with only finite-dimensional irreducible representations.

Abstract

We study boundaries for unital operator algebras. These are sets of irreducible $*$-representations that completely capture the spatial norm attainment for a given subalgebra. Classically, the Choquet boundary is the minimal boundary of a function algebra and it coincides with the collection of peak points. We investigate the question of minimality for the non-commutative counterpart of the Choquet boundary and show that minimality is equivalent to what we call the Bishop property. Not every operator algebra has the Bishop property, but we exhibit classes of examples that do. Throughout our analysis, we exploit various non-commutative notions of peak points for an operator algebra. When specialized to the setting of $C^*$-algebras, our techniques allow us to provide a new proof of a recent characterization of those $C^*$-algebras admitting only finite-dimensional irreducible representations.