Model-theoretic aspects of the Gurarij operator system

TL;DR

This paper explores the model-theoretic properties of the Gurarij operator system, establishing its uniqueness, primeness, and elementary embeddings, and examining the model theory of related finite-dimensional spaces.

Contribution

It proves the Gurarij operator system is the unique separable 1-exact existentially closed and nuclear model, and shows the existence of model companions for finite-dimensional M_q-spaces and M_q-systems.

Findings

Gurarij operator system is the unique separable 1-exact existentially closed operator system.

It is the prime model of its theory.

No C*-algebra can be existentially closed as an operator system.

Abstract

We establish some of the basic model theoretic facts about the Gurarij operator system $\mathbb{GS}$ recently constructed by the second-named author. In particular, we show: (1) $\mathbb{GS}$ is the unique separable 1-exact existentially closed operator system; (2) $\mathbb{GS}$ is the unique separable nuclear model of its theory; (3) every embedding of $\mathbb{GS}$ into its ultrapower is elementary; (4) $\mathbb{GS}$ is the prime model of its theory; and (5) $\mathbb{GS}$ does not have quantifier-elimination, whence the theory of operator systems does not have a model companion. We also show that, for any $q\in \mathbb{N}$, the theories of $M_q$-spaces and $M_q$-systems do have a model companion, namely the Fra\"{i}ss\'{e} limit of the class of finite-dimensional $M_q$-spaces and $M_q$-systems respectively; moreover we show that the model companion is separably categorical. We conclude the paper by showing that no C$^*$ algebra can be existentially closed as an operator system.