In this article, the bi-Lipschitz embeddability of the sequence of countably branching diamond graphs $(D_k^\omega)_{k\in\mathbb{N}}$ is investigated. In particular it is shown that for every $\varepsilon>0$ and $k\in\mathbb{N}$, $D_k^\omega$ embeds bi-Lipschiztly with distortion at most $6(1+\varepsilon)$ into any reflexive Banach space with an unconditional asymptotic structure that does not admit an equivalent asymptotically uniformly convex norm. On the other hand it is shown that the sequence $(D_k^\omega)_{k\in\mathbb{N}}$ does not admit an equi-bi-Lipschitz embedding into any Banach space that has an equivalent asymptotically midpoint uniformly convex norm. Combining these two results one obtains a metric characterization in terms of graph preclusion of the class of asymptotically uniformly convexifiable spaces, within the class of separable reflexive Banach spaces with an unconditional asymptotic structure. Applications to bi-Lipschitz embeddability into $L_p$-spaces and to some problems in renorming theory are also discussed.Comment: 44 pages, 3 tables, 2 figure

Abstract. We show that a Banach space with a normalized symmetric basis behaving like that of p (p > 2) cannot coarsely embed into a Hilbert space.A (not necessarily continuous) map f between two metric spaces (X, d) and, improving a theorem due to A. N. Dranishnikov, G. Gong, V. Lafforgue, and G. Yu [DGLY], gave a characterization of coarse embeddability of general metric spaces into a Hilbert space using a result of Schoenberg on negative definite kernels. He used this characterization to show that the spaces L p (µ) coarsely embed into a Hilbert space for p < 2. In this article, we show that p does not coarsely embed into a Hilbert space when p > 2. It was already proved in [DGLY] that the Lipschitz universal space c 0 (see [A]) does not coarsely embed into a Hilbert space.In its full generality, the statement of our result is as follows: Theorem 1. Suppose that a Banach space X has a normalized symmetric basis(e n ) n and that lim infHilbert space.In [Y], Yu proved that a discrete metric space with bounded geometry must satisfy the coarse geometric Novikov conjecture if it coarsely embeds into a Hilbert space, and in [KY] G. Kasparov and Yu proved that to get the same conclusion it is sufficient that the metric space coarsely embeds into a uniformly convex Banach space. Our theorem suggests that the result of [KY] cannot be deduced from the earlier theorem in [Y], but as yet there is no example of a discrete metric space with bounded geometry which coarsely embeds into p for some 2 < p < ∞ but not

In 1999, Bates, Johnson, Lindenstrauss, Preiss and Schechtman asked whether a Banach space that is a uniform quotient of ℓp, 1 < p = 2 < ∞, must be isomorphic to a linear quotient of ℓp. We apply the geometric property (β) of Rolewicz to the study of uniform and Lipschitz quotient maps, and answer the above question positively for the case 1 < p < 2. We also give a necessary condition for a Banach space to have c0 as a uniform quotient.2010 Mathematics Subject Classification. 46B80, 46B25, 46T99.

Abstract. We show that a quasi-Banach space coarsely embeds into a Hilbert space if and only if it is isomorphic to a linear subspace of L 0 (µ) for some probability space (Ω, B, µ).A (not necessarily continuous) map f between two metric spaces (X, d) andIt was proved in [JR] that p does not coarsely embed into a Hilbert space when p > 2. The present article is a strengthening of that result by giving a full characterization of quasi-Banach spaces that coarsely embed into a Hilbert space. This result, as well as its proof, mirrors the theorem in [AMM] which characterizes spaces that uniformly embed into a Hilbert space. The combination of Theorem 1 and the theorem in [AMM] yield that a quasi-Banach space uniformly embeds into a Hilbert space if and only if it coarsely embeds into a Hilbert space. This is counterintuitive in that a uniform embedding gives information only on small distances while a coarse embedding gives information only on large distances. Theorem 1. A quasi-Banach space X coarsely embeds into a Hilbert space if and only if there is a probability space (Ω, B, µ) such that X is linearly isomorphic to a subspace of L 0 (µ).To prove Theorem 1, we use a result essentially contained in [JR] as is recalled in the following proposition.