This paper is devoted to investigating the Lévy-Milman concentration phenomenon of 1-Lipschitz maps from mm-spaces (metric measure spaces) to infinite dimensional metric spaces. Here, an mm-space is a triple (X, dX , µ X ), where dX is a complete separable metric on a set X and µ X a finite Borel measure on (X, dX ). The theory of concentration of 1-Lipschitz functions was first introduced by V. D. Milman in his investigation of asymptotic geometric analysis ( [17], [18], [19]). Nowadays, the theory blend with various areas of mathematics, such as geometry, functional analysis and infinite dimensional integration, discrete mathematics and complexity theory, probability theory, and so on (see [16], [21], [22], [24] and the references therein for further information).
The theory of concentration of maps into general metric spaces was first studied by M. Gromov ([11], [12], [13]). He established the theory by introducing the observable diameter ObsDiam Y (X; -κ) for an mm-space X, a metric space Y , and κ > 0 in [13] (see Section 2 for the definition of the observable diameter). Given a sequence {X n } ∞ n=1 of mm-spaces and a metric space Y , we note that lim n→∞ ObsDiam Y (X n ; -κ) = 0 for any κ > 0 if and only if for any sequence {f n : X n → Y } ∞ n=1 of 1-Lipschitz maps, there exists a sequence {m fn } ∞ n=1 of points in Y such that lim
for any ε > 0. If lim n→∞ ObsDiam R (X n ; -κ) = 0 for any κ > 0, then the sequence {X n } ∞ n=1 of mm-spaces is called a Lévy family. The Lévy families were first introduced and analyzed by Gromov and Milman in [10]. In our previous works [2], [3], [4], [5], the author proved that if a metric space Y is either an R-tree, a doubling space, a metric graph, or a Hadamard manifold, then lim n→∞ ObsDiam Y (X n ; -κ) = 0 holds for any κ > 0 and any Lévy family {X n } ∞ n=1 . To prove these results, we needed to assume the finiteness of the dimension of the target metric spaces.
In this paper, we treat the case where the dimension of the target metric space Y is infinite. The author has proved in [1] that if the target space Y is so big that an mmspace X with some homogeneity property can isometrically be embedded into Y , then its observable diameter ObsDiam Y (X; -κ) is not close to zero. It seems from this result that the concentration to an infinite dimensional metric space cannot happen easily.
A main theorem of this paper is the following. For 1 ≤ p ≤ +∞, we denote by
n=1 be a sequence of mm-spaces and 1 ≤ p < q ≤ +∞. Then, the sequence {X n } ∞ n=1 is a Lévy family if and only if
As a result, we obtain the example of the infinite dimensional target metric space such that the concentration to the space happens as often as the concentration to the real line.
The proof of the sufficiency of Theorem 1.1 is easy. A. Gournay and M. Tsukamoto’s observations play important roles for the proof of the converse ( [9], [28]). Answering a question of Gromov in [14, Section 1.1.4], Tsukamoto proved in [28] that the “macroscopic” dimension of the space (B ∞ ℓ p , dℓ q ) for 1 ≤ p < q ≤ +∞ is finite. Gournay independently proved it in [9] in the case of q = +∞. For any p and q with 1 ≤ q ≤ p ≤ +∞, we have an example of a Lévy family which does not satisfy (1.1) (see Proposition 4.4).
As applications of Theorem 1.1, by virtue of [3,Propositions 4.3 and 4.4], we obtain the following corollaries of a Lévy group action. A Lévy group was first introduced by Gromov and Milman in [10]. Let a topological group G acts on a metric space X. The action is called bounded if for any ε > 0 there exists a neighborhood U of the identity element e G ∈ G such that dX (x, gx) < ε for any g ∈ U and x ∈ X. Note that every bounded action is continuous. We say that the topological group G acts on X by uniform isomorphisms if for each g ∈ G, the map X ∋ x → gx ∈ X is uniform continuous. The action is said to be uniformly equicontinuous if for any ε > 0 there exists δ > 0 such that dX (gx, gy) < ε for every g ∈ G and x, y ∈ X with dX (x, y) < δ. Given a subset S ⊆ G and x ∈ X, we put Sx := {gx | g ∈ S}.
Corollary 1.2. Let 1 ≤ p < q ≤ +∞ and assume that a Lévy group G boundedly acts on the metric space (B ∞ ℓ p , dℓ q ) by uniform isomorphisms. Then for any compact subset K ⊆ G and any ε > 0, there exists a point
There are no non-trivial bounded uniformly equicontinuous actions of a Lévy group to the metric space (B ∞ ℓ p , dℓ q ) for 1 ≤ p < q ≤ +∞. Gromov and Milman pointed out in [10] that the unitary group U(ℓ 2 ) of the separable Hilbert space ℓ 2 with the strong topology is a Lévy group. Many concrete examples of Lévy groups are known by the works of S. Glasner [8], H. Furstenberg and B. Weiss (unpublished), T. Giordano and V. Pestov [6], [7], and Pestov [25], [26]. For examples, groups of measurable maps from the standard Lebesgue measure space to compact groups, unitary groups of some von Neumann algebras, groups of measure and measure-class preserving automorphisms of the standard Lebesgue meas
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