Complete metric spaces with property (Z) are length metric spaces

arXiv:1808.00715v1 [math.FA] 2 Aug 2018

COMPLETE METRIC SPACES WITH PROPERTY (Z) ARE LENGTH SPACES ´ AND GONZALO MART´INEZ-CERVANTES ANTONIO AVILES Abstract. We prove that every complete metric space with property (Z) is a length space. These answers questions posed by Garc´ıa-Lirola, Proch´azka and Rueda Zoca, and by Becerra Guerrero, L´ opez-P´erez and Rueda Zoca, related to the structure of Lipschitzfree Banach spaces of metric spaces.

1. Introduction Let M be a metric space with a distinguished point 0 ∈ M. The set Lip0 (M) = {f : M → R Lipschitz : f (0) = 0} is a Banach space when endowed with the norm |f (x) − f (x)| . kf k = sup d(x, y) x6=y It turns out that Lip0 (M) is always a dual space and that the space generated by all functionals of the form δm : Lip0 (M) → R with δm (f ) = f (m) for every f ∈ Lip0 (M) is a predual which is usually denoted by F (M) and it is called the Lipschitz-free space of M. During the last decades the relations between metric properties of M and geometrical properties of F (M) have been deeply studied. In this paper we are going to focus on the following two properties (we denote the maximum of two real numbers a and b by a ∨ b and its minimum by a ∧ b): Definition 1.1. A pair (x, y) of points of M with x 6= y is said to have property (Z) if for every ε > 0 there exists z ∈ M \ {x, y} such that d(x, z) + d(z, y) ≤ d(x, y) + ε · (d(x, z) ∧ d(z, y)). M is said to have property (Z) if each pair of distinct points of M has property (Z). Definition 1.2. A complete metric space M is said to be a length space if for every x, y ∈ M and every δ > 0 there exists z ∈ M such that d(x, z) ∨ d(z, y) ≤ d(x, y)/2 + δ. 2010 Mathematics Subject Classification. 46B20,54E50. Key words and phrases. length metric space, Lipschitz-free space, Daugavet property, diameter-2 property, strongly exposed points, preserved extreme points. Authors supported by projects MTM2014-54182-P and MTM2017-86182-P (Government of Spain, AEI/FEDER, EU) and by project 19275/PI/14 (Fundaci´on S´eneca).

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Property (Z) was introduced in [6] in order to characterize local metric spaces in the compact case. For a complete metric space being local is equivalent to being length [5, Proposition 2.4]. Papers [6] and [5] are devoted to the study of spaces of Lipschitz functions with the Daugavet property. Recall that a Banach space X is said to have the Daugavet property if kI + T k = 1 + kT k for every rank-one operator T : X → X, where I : X → X denotes the identity operator. The Daugavet property for Lipschitz-free spaces is characterized as follows: Theorem 1.3 ([5],[6]). Let M be a complete metric space. F (M) has the Daugavet property if and only if Lip0 (M) has the Daugavet property if and only if M is length. It can be easily proved that every length space has property (Z). On the other hand, it was proved in [6] that both properties are equivalent in compact metric spaces. The importance of Property (Z) in the study of Lipschitz-free spaces can be inferred from the following Theorem: Theorem 1.4 ([5]). Let M be a complete metric space. Then the following assertions are equivalent: (1) (2) (3) (4)

M has property (Z); The unit ball of F (M) does not have strongly exposed points; The norm of Lip0 (M) does not have any point of Gˆateaux differentiability; The norm of Lip0 (M) does not have any point of Fr´echet differentiability.

For more relations among property (Z), being length and the extremal structure of Lipschitz-free spaces we refer the reader to [2] and [4]. These results motivated the following question posed in [5, Question 1]: Question. If M is a complete metric space with property (Z), is M length? In this paper we provide an affirmative answer to this question: Main Theorem. A complete metric space M is length if and only if it has property (Z). As a consequence, all conditions appearing in Theorems 1.3 and 1.4 are equivalent. Moreover, [5, Proposition 4.9] asserts that if M is a complete metric length space, then the unit ball of F (M) does not have preserved extreme points. Notice that every strongly exposed point is a preserved extreme point and that in Lipschitz-free spaces there might be preserved extreme points which are not strongly exposed [4, Example 6.4]. Nevertheless, it follows from the aforementioned results and our Main Theorem that, for Lipschitz-free spaces, the absence of preserved extreme points in the unit ball is equivalent to the absence of strongly exposed points in the unit ball. Furthermore, every Banach space with the Daugavet property has the strong diameter 2 property [1, Theorem 4.4] and a simple computation shows that the unit ball of a Banach space with the slice diameter 2 property cannot contain strongly exposed points (see [3] for

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definitions). Thus, the Main Theorem also provides a complete answer for the scalar case of [3, Question 3.3]. Let us summarize in one theorem all the equivalences obtained: Theorem 1.5. Let M be a complete metric space. Then the following assertions are equivalent: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

M has property (Z); M is length; The unit ball of F (M) does not have strongly exposed points; The unit ball of F (M) does not have preserved extreme points; F (M) has the Daugavet property; Lip0 (M) has the Daugavet property; The norm of Lip0 (M) does not have any point of Gˆateaux differentiability; The norm of Lip0 (M) does not have any point of Fr´echet differentiability; F (M) has the slice diameter 2 property; F (M) has the diameter 2 property; F (M) has the strong diameter 2 property. 2. Preliminaries

In this section we state transfinite versions of several basic facts about sequences in metric spaces. Definition 2.1. If α < ω1 is a limit ordinal and {xγ : γ < α} is a transfinite sequence of points of a metric space M, we say that the sequence is Cauchy if for every ε > 0 there exists β < α such that d(xγ , xδ ) < ε whenever β < γ < δ < α. Similarly, we say that {xγ : γ < α} converges to x if for every ε > 0 there exists β < α such that d(xγ , x) < ε whenever β < γ < α. Lemma 2.2. A sequence {xγ : γ < α} is Cauchy (respectively, convergent to a point x) if and only if {xαn : n ∈ N} is Cauchy (respectively, convergent to x) whenever α1 < α2 < · · · and supn αn = α. As an immediate consequence of the above, in a complete metric space every transfinite Cauchy sequence is convergent. Lemma 2.3. Let α be a limit ordinal and {xγ : γ < α} be a sequence of points in a complete metric space M such that {xγ : γ < β} converges to xβ for every limit ordinal β < α. Suppose that X d(xγ , xγ+1 ) < +∞. γ