Lipschitz free spaces isomorphic to their infinite sums and geometric applications

Abstract

We find general conditions under which Lipschitz-free spaces over metric spaces are isomorphic to their infinite direct 1sum and exhibit several applications. As examples of such applications we have that Lipschitz-free spaces over balls and spheres of the same finite dimensions are isomorphic, that the Lipschitz-free space over Z d is isomorphic to its 1 -sum, or that the Lipschitz-free space over any snowflake of a doubling metric space is isomorphic to 1 . Moreover, following new ideas of Bruè et al. from [10] we provide an elementary self-contained proof that Lipschitz-free spaces over doubling metric spaces are complemented in Lipschitz-free spaces over their superspaces and they have BAP. Everything, including the results about doubling metric spaces, is explored in the more comprehensive setting of p-Banach spaces, which allows us to appreciate the similarities and differences of the theory between the cases p < 1 and p = 1.

Cantor sets of low density and Lipschitz functions on $C^1$ curves

“…Moreover, this line of research recently has gained popularity due to the importance that Lipschitz maps have in the theory of nonlinear geometry of Banach spaces. Furthermore, the study of Lipschitz-free spaces, which are isometric preduals of the Lipschitz spaces, is a very active line of research nowadays (see [1], [2], [5], [6], and [8] for instance).…”

Lipschitz algebras and Lipschitz-free spaces over unbounded metric spaces

“…By (2.5), to address this question it suffices to consider the case when M is bounded. We recall that E p,1,M is always one-to-one on P(M), but this does not guarantee its injectivity on its completion F p (M) (see [5] for a discussion about the injectivity of envelope maps).…”

“…Indeed, if M does not have isolated points we add an isolated point * to our space M so that F p (M ∪ { * }) ≃ F p (M) for every 0 < p ≤ 1 (see e.g. [4,Lemma 2.8]). This way we obtain a homeomorphism between M and B := B(M ∪ { * }, α) \ {0} such that F p (M) ≃ F p (B).…”

On large $\ell_1$-sums of Lipschitz-free spaces and applications

“…Ideas and techniques developed years ago by N. J. Kalton [12] have proven to be invaluable in this field, see for example [1], [2], [4], [13]. By applying some of Kalton's results [12, §4], P. L. Kauffman [13] established the striking fact that, for all Banach spaces X, F(X) is linearly isomorphic to its ℓ 1 -sum.…”

“…Furthermore, for all Banach spaces X, F(X) is linearly isomorphic to F(B X ), where B X denotes the unit ball of X. These theorems are now part of the basic toolkit in any study of this kind and we refer to [1] for a deep investigation on this topic.…”