The Bishop–Phelps–Bollobás property for Lipschitz maps

Abstract

In this paper, we introduce and study a Lipschitz version of the Bishop-Phelps-Bollobás property (Lip-BPB property). This property deals with the possibility of making a uniformly simultaneous approximation of a Lipschitz map F and a pair of points at which F almost attains its norm by a Lipschitz map G and a pair of points such that G strongly attains its norm at the new pair of points. We first show that if M is a finite pointed metric space and Y is a finite-dimensional Banach space, then the pair (M, Y ) has the Lip-BPB property, and that both finiteness assumptions are needed. Next, we show that if M is a uniformly Gromov concave pointed metric space (i.e. the molecules of M form a set of uniformly strongly exposed points), then (M, Y ) has the Lip-BPB property for every Banach space Y . We further prove that this is the case for finite concave metric spaces, ultrametric spaces, and Hölder metric spaces. The extension of the Lip-BPB property from (M, R) to some Banach spaces Y and some results for compact Lipschitz maps are also discussed.Recently, the problem of deciding for which metric spaces M the set LipSNA(M, Y ) is (norm) dense in Lip 0 (M, Y ) has been studied. We refer the reader to [11], [16], [18], and [21], as references for this study. Let us collect here some of the known results. First, the density does not always hold, as in [21, Example 2.1] it is shown that LipSNA([0, 1], R) is not dense in Lip 0 ([0, 1], R). In fact, this can be generalized to length spaces [11, Theorem 2.2]. On the other hand, it is obvious that LipSNA(M, Y ) = Lip 0 (M, Y ) when M is finite (actually, this fact characterizes finiteness of M ). Besides, LipSNA(M, Y ) is dense in Lip 0 (M, Y ) for every Banach space Y when M is uniformly discrete, or M is countable and compact, or M is a compact Hölder metric space (i.e. M = (N, d θ ) for some metric space (N, d) and 0 < θ < 1), but

Some stability properties for the Bishop–Phelps–Bollobás property for Lipschitz maps

“…Hence, we are studying for which metric spaces M and Banach spaces Y the set of those bounded linear operators from F(M ) to Y which attain their norm at some molecule is dense in L(F(M ), Y ). Furthermore, we are also interested in the following stronger version of density, which was introduced in [7].…”

“…On the other hand, the converse is far from being true. In fact, if M is a finite pointed metric space, while it is clear that LipSNA(M, Y ) = Lip 0 (M, Y ) for every Banach space Y , Example 2.5 in [7] shows that one can find finite pointed metric spaces M and Banach spaces Y such that (M, Y ) fails to have the Lip-BPB property. For this reason, throughout this paper each of these notions of approximation by strongly norm attaining Lipschitz maps will be discussed separately.…”

Emerging notions of norm attainment for Lipschitz maps between Banach spaces

“…If v n / ∈ supp F , then we must have that w n ∈ supp F by (8). Hence, we have dist(v n , [x, y]) v n − w n + dist(w n , [x, y]) < ρ(x, y), which completes the proof.…”

“…However, concerning the problem of the denseness of norm attaining Lipschitz maps, it is impossible to proceed further with this definition. In fact, SNA(X, R) fails to be dense in Lip 0 (X, R) for every Banach space X (see [16,Theorem 2.3]) and, therefore, SNA(X, Y ) cannot be dense in Lip 0 (X, Y ) for any Banach space Y by [8,Proposition 4.2]. We refer the interested reader to the recent papers [6,8] for the study of the denseness of strongly norm attaining Lipschitz maps defined in general metric spaces.…”

Norm-attaining lattice homomorphisms

“…In fact, this theory has been widely extended to different contexts besides linear functionals. Indeed, among others, some authors considered it in the context of linear operators (see [13,26,28,31,37,39,40]); others studied norm-attaining bilinear mappings (see [4,8,18]); and more recently several problems on norm-attainment of homogeneous polynomials and Lipschitz maps were considered (see [5,9] and [15,16,17], respectively). In the context of homomorphisms on Banach lattices, we should highlight the recent paper [34], where a James type theorem was proved for positive linear functionals on some Banach lattices (see [34, §6]).…”