An approximate unique extension property for completely positive maps

TL;DR

This paper investigates the structure of certain $ ext{C}^*$-representations related to non-commutative Choquet boundaries, revealing restrictions on their extensions and connecting to major conjectures in operator algebra theory.

Contribution

It introduces an approximate unique extension property for completely positive maps and links this to Arveson's hyperrigidity conjecture and classical operator theory conjectures.

Findings

Characterizes the closure of the unitary orbit in the non-commutative Choquet boundary.

Shows restrictions on unital completely positive extensions of approximately unitarily equivalent $ ext{C}^*$-representations.

Reformulates the Šaškin Theorem and Arveson's essential normality conjecture.

Abstract

We study the closure of the unitary orbit of a given point in the non-commutative Choquet boundary of a unital operator space with respect to the topology of pointwise norm convergence. This may be described more extensively as the $\ast$-representations of the $\mathrm{C}^{\ast}$-envelope that are approximately unitarily equivalent to one that possesses the unique extension property. Although these $\ast$-representations do not necessarily have the unique extension property themselves, we show that their unital completely positive extensions display significant restrictions. When the underlying operator space is separable, this allows us to connect our work to Arveson's hyperrigidity conjecture. Finally, as an application, we reformulate the classical \v{S}a\v{s}kin Theorem and Arveson's essential normality conjecture.