Corrigendum to: Essential normality, essential norms and hyperrigidity

TL;DR

This corrigendum clarifies the conditions under which certain operator system properties hold, specifically providing additional justification for previous claims about essential normality and hyperrigidity related to polynomial ideals.

Contribution

The paper offers a corrected proof for the unique extension property of operator systems derived from polynomial ideals, under specific assumptions on the ideals.

Findings

Proposition 4.11 and Theorem 4.12 are valid under new assumptions.

Homogeneous ideals that are sufficiently non-trivial satisfy these assumptions.

The general case remains unresolved.

Abstract

In our paper "Essential normality, essential norms and hyperrigidity" we claimed that the restriction of the identity representation of a certain operator system (constructed from a polynomial ideal) has the unique extension property, however the justification we gave was insufficient. In this note we provide the required justification under some additional assumptions. Fortunately, homogeneous ideals that are "sufficiently non-trivial" are covered by these assumptions. This affects the section of our paper relating essential normality and hyperrigidity. We show here that Proposition 4.11 and Theorem 4.12 hold under the additional assumptions. We do not know if they hold in the generality considered in our paper.