$C^*$-supports and abnormalities of operator systems

TL;DR

This paper investigates the conditions under which $C^*$-supports of operator systems are unique, and explores how these supports relate to properties like hyperrigidity and abnormalities, advancing the understanding of operator system extensions.

Contribution

It establishes criteria for the uniqueness of $C^*$-supports using Hamana's theory and links this to the extension property and hyperrigidity of operator systems.

Findings

Uniqueness of $C^*$-supports characterized by containment in injective envelopes

New characterizations of the extension property for $*$-representations

Description of abnormalities via collections of $C^*$-supports

Abstract

Let $S$ be a concrete operator system represented on some Hilbert space $H$. A $C^*$-support of $S$ is the $C^*$-algebra generated (via the Choi--Effros product) by $S$ inside an injective operator system acting on $H$. By leveraging Hamana's theory, we show that such a $C^*$-support is unique precisely when $C^*(S)$ is contained in every copy of the injective envelope of $S$ that acts on $H$. Further, we demonstrate how the uniqueness of certain $C^*$-supports can be used to give new characterizations of the unique extension property for $*$-representations, as well as the hyperrigidity of $S$. In another direction, we utilize the collection of all $C^*$-supports of $S$ to describe the subspace generated by the so-called abnormalities of $S$, thereby complementing a result of Kakariadis.