Model-theoretic aspects of the Gurarij operator system

2015/01/18 by Isaac Goldbring, Martino Lupini, Goldbring, Isaac +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Logic (math.LO) #Operator Algebras (math.OA) #math.FA #math.LO #math.OA

paper · pdf · doi:10.48550/arxiv.1501.04332

20 pages; major changes in statements of the main results

arxiv created 2015/04/28 · arxiv updated 2015/04/29

Abstract

We establish some of the basic model theoretic facts about the Gurarij operator system \mathbbGS recently constructed by the second-named author. In particular, we show: (1) \mathbbGS is the unique separable 1-exact existentially closed operator system; (2) \mathbbGS is the unique separable nuclear model of its theory; (3) every embedding of \mathbbGS into its ultrapower is elementary; (4) \mathbbGS is the prime model of its theory; and (5) \mathbbGS does not have quantifier-elimination, whence the theory of operator systems does not have a model companion. We also show that, for any q∈ ℕ, the theories of M_q-spaces and M_q-systems do have a model companion, namely the Fraïssé 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.

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