Omitting types in operator systems

TL;DR

This paper explores the definability and extension properties of operator systems and C*-algebras, revealing limitations of uniform definability, establishing equivalences of key properties, and connecting these to the Kirchberg Embedding Problem.

Contribution

It demonstrates that 1-exact operator systems are not uniformly definable, links WEP to existential closedness, and introduces a new space related to the Kirchberg Embedding Problem.

Findings

1-exact operator systems are not uniformly definable by types

WEP is equivalent to existential closedness and the complete tight Riesz interpolation property

A new space of operator systems relates to the Kirchberg Embedding Problem

Abstract

We show that the class of 1-exact operator systems is not uniformly definable by a sequence of types. We use this fact to show that there is no finitary version of Arveson's extension theorem. Next, we show that WEP is equivalent to a certain notion of existential closedness for C$^*$ algebras and use this equivalence to give a simpler proof of Kavruk's result that WEP is equivalent to the complete tight Riesz interpolation property. We then introduce a variant of the space of n-dimensional operator systems and connect this new space to the Kirchberg Embedding Problem, which asks whether every C$^*$ algebra embeds into an ultrapower of the Cuntz algebra $\mathcal{O}_2$. We end with some results concerning the question of whether or not the local lifting property (in the sense of Kirchberg) is uniformly definable by a sequence of types in the language of C$^*$ algebras.