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

Abstract

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...

Amalgamation and Ramsey properties of $L_p$ spaces

“…Example 5.5. The classes { n ∞ } n , the finite dimensional polyhedral spaces, and all finite dimensional normed spaces have the (SARP + ) (proved by D. Bartošová, M. Lupini, B. Mbombo and the second author of this paper, in [BaLALuMbo2] (see also [BaLALuMbo1]).…”

“…The theory of abstract ultrahomogeneous metric structures has been studied in [BYBeHeUs], while the KPT correspondence has been recently extended to this context by J. Melleray and T. Tsankov [MeTsa]. The KPT correspondence for metric structures was for the first time used in [BaLALuMbo1], [BaLALuMbo2] in showing, for example, that the isometry group of the Gurarij space is extremely amenable by supplying an appropriate Ramsey type result which relies on the Graham-Rothschild Theorem. We note that Gurarij space shares this this property with the infinite dimensional Hilbert spaces and the spaces L p (0, 1), proved by M. Gromov and V. D. Milman [GrMi], and by T. Giordano and V. Pestov [GiPe] respectively, relying on the method of concentration of measure.…”

“…In general, a class of finite dimensional G has the (ARP) when for every X, Y ∈ G and every ε > 0 there is Z ∈ G such that for every 1-Lipschitz mapping c : Emb(X, Z) → [0, 1] there is some isometric embedding γ : Y → Z such that the oscillation of c in the set of compositions γ •Emb(X, Y ) is at most ε. It is interesting to mention that, while the proof of the (ARP) of { n ∞ } n uses the dual Ramsey Theorem (DR) of Graham and Rothschild (see [BaLALuMbo1] ), our proof of the (ARP) of { n p } n p = 2, ∞ utilize an arithmetical version of (DR), namely, that for partitions of equal sized pieces.…”

The Ramsey Properties for Grassmannians Over ℝ, ℂ

“…that AgepL p r0, 1sq does not have the pSARP `q because in these spaces there are finite dimensional isometric subspaces, one well complemented and the other badly complemented. ‚ F " AgepCr0, 1sq: This is proved in [4] (see also [2]), directly using injective envelopes and some approximations, or as a byproduct of the pSARP `q of the family tℓ n 8 u n and the Kechris-Pestov-Todorcevic correspondence for Banach spaces.…”

“…The case 1 ď p ‰ 2 ă 8 follows from the approximate Ramsey property of this class, proved in [6] and the result of G. Schechtman in [17] stating that tℓ n p u n are amalgamation classes. The case p " 8 is proved in [4] (see also [2]) using the dual Ramsey Theorem. ‚ AgepL p r0, 1sq for p R 2N `4: This is a byproduct of the extreme amenability of IsopL p r0, 1sq, proved by T. Giordano and V. Pestov in [7], the (KPT) correspondence, and the fact that AgepL p r0, 1sq is an amalgamation class, proved in [6].…”