Omitting types in operator systems

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.

25 pages; final version to appear in Indiana University Mathematics Journal; significant clarification of the exposition and a couple new results, including the fact that LLP is equivalent to the local matrix ultraproduct lifting property