Operator space and operator system analogs of Kirchberg's nuclear embedding theorem

Advances in Mathematics

2018

Self Cite

We provide a unified approach to Fraïssé limits in functional analysis, including the Gurarij space, the Poulsen simplex, and their noncommutative analogs. We obtain in this general framework many known and new results about the Gurarij space and the Poulsen simplex, and at the same time establish their noncommutative analogs. Particularly, we construct noncommutative analogs of universal operators in the sense of Rota.2000 Mathematics Subject Classification. Primary 46L07, 46A55; Secondary 46L89, 03C30, 03C98.

Section: More Examplessupporting

confidence: 66%

Fraïssé limits in functional analysis

Advances in Mathematics

2018

Self Cite

“…The operator spaces G q . The following amalgamation result is proved in [23,Lemma 3.1]; see also [25, Lemma 2.1]. Lemma 3.1.…”

Section: 2mentioning

confidence: 97%

“…Precise uniqueness of the Gurarij operator space was later proven in [23] by realizing the Gurarij operator space (henceforth referred to as NG) as the Fraïssé limit of the class of finite-dimensional 1-exact operator spaces. In [24], the second author established the existence and uniqueness of the Guarij operator system GS, which is the Fraïsé limit Goldbring's work was partially supported by NSF CAREER grant DMS-1349399.…”

Section: Introductionmentioning

confidence: 99%

Model-theoretic aspects of the Gurarij operator system

Goldbring

2018

Isr. J. Math.

Self Cite

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 Fraïssé 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.

“…The function systems approach has been adopted in the work of Conley and Törnquist [13] and, independently, in [45,46], where it is shown that the class of finite-dimensional function systems is a Fraïssé class. Its limit can be identified with the function system A(P) corresponding to the Poulsen simplex, which we will call the Poulsen system.…”

Section: 1mentioning

confidence: 99%

The Ramsey property for Banach spaces and Choquet simplices, and applications

Bartošová

Lopez-Abad

Comptes Rendus. Mathématique

et al. 2017

Self Cite

Abstract. We show that the Gurarij space G and its noncommutative analog NG both have extremely amenable automorphism group. We also compute the universal minimal flows of the automorphism groups of the Poulsen simplex P and its noncommutative analogue NP. The former is P itself, and the latter is the state space of the operator system associated with NP. This answers a question of Conley and Törnquist. We also show that the pointwise stabilizer of any closed proper face of P is extremely amenable. Similarly, the pointwise stabilizer of any closed proper biface of the unit ball of the dual of the Gurarij space (the Lusky simplex) is extremely amenable.These results are obtained via the Kechris-Pestov-Todorcevic correspondence, by establishing the approximate Ramsey property for several classes of finite-dimensional operator spaces and operator systems (with distinguished linear functionals), including: Banach spaces, exact operator spaces, function systems with a distinguished state, and exact operator systems with a distinguished state. This is the first direct application of the Kechris-Pestov-Todorcevic correspondence in the setting of metric structures. The fundamental combinatorial principle that underpins the proofs is the Dual Ramsey Theorem of Graham and Rothschild.In the second part of the paper, we obtain factorization theorems for colorings of matrices and Grassmannians over R and C, which can be considered as continuous versions of the Dual Ramsey Theorem for Boolean matrices and of the Graham-Leeb-Rothschild Theorem for Grassmannians over a finite field.Recall that given a topological group G, a compact G-space or G-flow is a compact Hausdorff space X endowed with a continuous action of G. Such a G-flow X is called minimal when every orbit is dense. There is a natural notion of morphism between G-flows, given by a (necessarily surjective) G-equivariant continuous map (factor). A minimal G-flow is universal if it factors onto any other minimal G-flow. It is a classical fact that any topological group G admits a unique (up to isomorphism of G-flows) universal minimal flow, usually denoted by M (G) [18,33]. For any locally compact non compact Polish group G, the universal minimal G-flow is nonmetrizable. At the opposite end, non locally compact topological groups often have metrizable universal minimal flows, or even reduced to a single point. A topological group for which M (G) is a singleton is called extremely amenable. (Amenability of G is equivalent to the assertion that every compact G-space has an invariant Borel measure. Thus any extremely amenable group is, in particular, amenable.)The universal minimal flow has been explicitly computed for many topological groups, typically given as automorphism groups of naturally arising mathematical structures. Examples of extremely amenable Polish groups include the group of order automorphisms of Q [61], the group of unitary operators on the separable infinite-dimensional Hilbert space [30], the automorphism group of the hyperfinite II 1 factor and of in...