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Model-theoretic aspects of the Gurarij operator system

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Abstract

We establish some of the basic model theoretic facts about the Gurarij operator system GS recently constructed by the second-named author. In particular, we show: (1) GS is the unique separable 1-exact existentially closed operator system; (2) GS is the unique separable nuclear model of its theory; (3) every embedding of GS into its ultrapower is elementary; (4) GS is the prime model of its theory; and (5) 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 ∈ ℕ, the theories of Mq-spaces and Mq-systems do have a model companion, namely the Fra¨ıssé limit of the class of finite-dimensional Mq-spaces and Mq-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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Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Science and Engineering Offices M/C 249, 851 S. Morgan St., Chicago, IL, 60607-7045, USA

    Isaac Goldbring

  2. Department of Mathematics, University of California, Irvine, Irvine, CA, 92697-3875, USA

    Isaac Goldbring

  3. Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, Room 02.126, 1090, Wien, Austria

    Martino Lupini

  4. Mathematics Department, California Institute of Technology, 1200 East California Boulevard, Mail Code 253-37, Pasadena, CA, 91125, USA

    Martino Lupini

Authors
  1. Isaac Goldbring
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  2. Martino Lupini
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Correspondence to Isaac Goldbring.

Additional information

Goldbring’s work was partially supported by NSF CAREER grant DMS-1349399. Lupini’s work was supported by the York University Susan Mann Dissertation Scholarship and by the ERC Starting grant no. 259527 of Goulnara Arzhantseva. This work was initiated during a visit of the second author to the University of Illinois at Chicago. The hospitality of the UIC Mathematics Department is gratefully acknowledged.

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Goldbring, I., Lupini, M. Model-theoretic aspects of the Gurarij operator system. Isr. J. Math. 226, 87–118 (2018). https://doi.org/10.1007/s11856-018-1691-3

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  • DOI: https://doi.org/10.1007/s11856-018-1691-3

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