Reflexive representability and stable metrics

Abstract

Abstract. It is well-known that a topological group can be represented as a group of isometries of a reflexive Banach space if and only if its topology is induced by weakly almost periodic functions (see [Sht94], [Meg01b] and [Meg07]). We show that for a metrisable group this is equivalent to the property that its metric is uniformly equivalent to a stable metric in the sense of Krivine and Maurey (see [KM81]). This result is used to give a partial negative answer to a problem of Megrelishvili.

Equivariant Geometry of Banach Spaces and Topological Groups

“…Here a topological group G is said to admit a topologically faithful isometric linear representation on a reflexive Banach space E if G is isomorphic to a subgroup of the linear isometry group Isom(E) equipped with the strong operator topology. The following theorem combines results due to A. Shtern [61], M. Megrelishvili [41] and I. Ben Yaacov, A. Berenstein and S. Ferri [10]. Theorem 52.…”

“…For our next results, we define a topological group to be metrically stable if it admits a compatible left-invariant stable metric. By results of [10,41,61], a Polish group is metrically stable if and only if it is isomorphic to a subgroup if the linear isometry group of a separable reflexive Banach space under the strong operator topology. Also, by a result of Y. Raynaud [52], a metrically stable Banach space contains a copy of ℓ p for some 1 p < ∞.…”

“…Theorem 52. [61,41,10] For a Polish group G, the following are equivalent, (1) G is metrically stable, (2) G has a topologically faithful isometric linear representation on a separable reflexive Banach space, (3) for every identity neighbourhood V ⊆ G, there is a continuous weakly almost periodic function φ :…”

Model Theoretic Stability and Definability of Types, After A. Grothendieck

“…In a meeting in Kolkata in January 2013, the author asked the audience who it was, and when, to have first defined the notion of a stable formula, and to the expected answer replied that, no, it had been Grothendieck, in the fifties. This was meant as a joke, of course-a more exact statement would be that in Théorème 6 and Proposition 7 of Grothendieck [6] there appears a condition (see (1) and 2 below) which can be recognised as the "non order property" (NOP) 1 . It took (us) a while longer to realise that one could ask, quite seriously, who first proved the "Fundamental Theorem of Stability Theory", namely, the equivalence between NOP and definability of types, and the answer would essentially be the same.…”

Embedding a topological group into its WAP-compactification

“…This is a far reaching generalization of [22, Théorème 5.1] from which Theorem 2.4 follows if one uses the fact ( [19]) that every reflexively representable group can be uniformly embedded in a reflexive Banach space. Later on, A. Berenstein, I. Ben-Yaacov and the first named author [1] proved that every reflexively representable topological group whose topology is induced by an invariant metric admits an equivalent distance that is both invariant and stable. Using this fact, nonreflexive representability of c 0 could be derived directly from [22,Théorème 5.1].…”