Model-theoretic aspects of the Gurarij operator system

Goldbring, Isaac; Lupini, Martino

doi:10.1007/s11856-018-1691-3

Cited by 9 publications

(16 citation statements)

References 30 publications

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“…It is also proved in [28] that NP is the unique nontrivial metrizable noncommutative Choquet simplex with dense matrix extreme boundary, and A(NP) is the unique separable nuclear operator systems that is universal in the sense of Kirchberg and Wassermann [63]. The model-theoretic properties of the noncommutative Poulsen system have been investigated in [49]. The following noncommutative analog of Theorem 1.3 follows from our general results; see Subsection 8.2.…”

Section: Introductionmentioning

confidence: 60%

Fraïssé limits in functional analysis

Lupini

2018

Advances in Mathematics

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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.

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Section: Introductionmentioning

confidence: 60%

Fraïssé limits in functional analysis

Lupini

2018

Advances in Mathematics

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“…, P (q) q ) of the corresponding limit A(P (q) ) is the q-minimal Poulsen simplex. The model-theoretic properties of A(P), A(NP), and A(P (q) ) have been studied in [25].…”

Section: 1mentioning

confidence: 99%

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

Bartošová

Lopez-Abad

Lupini

et al. 2017

Comptes Rendus. Mathématique

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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...

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“…Set K C * |L os := {A|L os : A ∈ K C * }. In [6], the following question was raised: is K C * |L os an elementary class? The main result of this note is to give an affirmative answer to this question.…”

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confidence: 99%

“…If K C * |L os were ∀∃-axiomatizable, then there would be A ∈ K C * |L os that is existentially closed for K C * |L os . However, in [6,Section 5], it was observed that if φ : X → Y is a complete order embedding that is also existential, then φ maps unitaries to unitaries. Take a complete order embedding of A into a C * -algebra B that is not a * -homomorphism (see, for example, [6, Section 5]); since this embedding maps unitaries to unitaries (since A is existentially closed for K), this contradicts Corollary 15.…”

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confidence: 99%