arXiv subjects
Publications and source records attributed to Marek Cuth.
We conjecture that whenever $M$ is a metric space of density at most continuum, then the space of Lipschitz functions is $w^*$-separable. We prove the conjecture for several classes of metric spaces including all the Banach spaces with a projectional skeleton, Banach spaces with a $w^*$-separable dual unit ball and locally separable complete metric spaces.
We present a way to turn an arbitrary (unbounded) metric space $\mathcal{M}$ into a bounded metric space $\mathcal{B}$ in such a way that the corresponding Lipschitz-free spaces $\mathcal{F}(\mathcal{M})$ and $\mathcal{F}(\mathcal{B})$ are isomorphic. The construction we provide is functorial in a weak sense and has the advantage of being explicit. Apart from its intrinsic theoretical interest, it has many applications in that it allows to transfer many arguments valid for Lipschitz-free spaces over bounded spaces to Lipschitz-free spaces over unbounded spaces. Furthermore, we show that with a slightly modified point-wise multiplication, the space $\rm{Lip}_0(\mathcal{M})$ of scalar-valued Lipschitz functions vanishing at zero over any (unbounded) pointed metric space is a Banach algebra with its canonical Lipschitz norm.
We find general conditions under which Lipschitz-free spaces over metric spaces are isomorphic to their infinite direct $\ell_1$-sum 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 $\mathbb{Z}^d$ is isomorphic to its $\ell_1$-sum, or that the Lipschitz-free space over any snowflake of a doubling metric space is isomorphic to $\ell_1$. Moreover, following new ideas from [E. Bruè, S. Di Marino and F. Stra, Linear Lipschitz and $C^1$ extension operators through random projection, arXiv:1801.07533] 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 $p=1$.
Our aim in this article is to contribute to the theory of Lipschitz free $p$-spaces for $0<p\le 1$ over the Euclidean spaces $\mathbb{R}^d$ and $\mathbb{Z}^d$. To that end, on one hand we show that $\mathcal{F}_p(\mathbb{R}^d)$ admits a Schauder basis for every $p\in(0,1]$, thus generalizing the corresponding result for the case $p=1$ achieved in [P. Hájek and E. Pernecká, On Schauder bases in Lipschitz-free spaces, J. Math. Anal. Appl. 416 (2014), no. 2, 629--646] and answering in the positive a question that was raised in [F. Albiac, J. L. Ansorena, M. Cúth, and M. Doucha, Embeddability of lp and bases in Lipschitz free $p$-spaces for $0 < p \le 1$, J. Funct. Anal. 278 (2020), no. 4, 108354, 33]. Explicit formulas for the bases of both $\mathcal{F}_p(\mathbb{R}^d)$ and its isomorphic space $\mathcal{F}_p([0,1]^d)$ are given. On the other hand we show that the well-known fact that $\mathcal{F}(\mathbb{Z})$ is isomorphic to $\ell_{1}$ does not extend to the case when $p<1$, that is, $\mathcal{F}_{p}(\mathbb{Z})$ is not isomorphic to $\ell_p$ when $0<p<1$.
Our goal in this paper is to continue the study initiated by the authors in [Lipschitz free $p$-spaces for $0<p<1$; arXiv:1811.01265 [math.FA]] of the geometry of the Lipschitz free $p$-spaces over quasimetric spaces for $0<p\le1$, denoted $\mathcal F_{p}(\mathcal M)$. Here we develop new techniques to show that, by analogy with the case $p=1$, the space $\ell_{p}$ embeds isomorphically in $\mathcal F_{p}(\mathcal M)$ for $0<p<1$. Going further we see that despite the fact that, unlike the case $p=1$, this embedding need not be complemented in general, complementability of $\ell_{p}$ in a Lipschitz free $p$-space can still be attained by imposing certain natural restrictions to $\mathcal M$. As a by-product of our discussion on basis in $\mathcal F_{p}([0,1])$, we obtain the first-known examples of $p$-Banach spaces for $p<1$ that possess a basis but fail to have an unconditional basis.
This paper initiates the study of the structure of a new class of $p$-Banach spaces, $0<p<1$, namely the Lipschitz free $p$-spaces (alternatively called Arens-Eells $p$-spaces) $\mathcal{F}_{p}(\mathcal{M})$ over $p$-metric spaces. We systematically develop the theory and show that some results hold as in the case of $p=1$, while some new interesting phenomena appear in the case $0<p<1$ which have no analogue in the classical setting. For the former, we, e.g., show that the Lipschitz free $p$-space over a separable ultrametric space is isomorphic to $\ell_{p}$ for all $0<p\le 1$, or that $\ell_p$ isomorphically embeds into $\mathcal{F}_p(\mathcal{M})$ for any $p$-metric space $\mathcal{M}$. On the other hand, solving a problem by the first author and N. Kalton, there are metric spaces $\mathcal{N}\subset \mathcal{M}$ such that the natural embedding from $\mathcal{F}_p(\mathcal{N})$ to $\mathcal{F}_p(\mathcal{M})$ is not an isometry.
We prove that the Lipschitz-free space over a separable ultrametric space has a monotone Schauder basis and is isomorphic to $\ell_1$. This extends results of A. Dalet using an alternative approach.
In this note we study the structure of Lipschitz-free Banach spaces. We show that every Lipschitz-free Banach space over an infinite metric space contains a complemented copy of $\ell_1$. This result has many consequences for the structure of Lipschitz-free Banach spaces. Moreover, we give an example of a countable compact metric space $K$ such that $F(K)$ is not isomorphic to a subspace of $L_1$ and we show that whenever $M$ is a subset of $R^n$, then $F(M)$ is weakly sequentially complete; in particular, $c_0$ does not embed into $F(M)$.
We study a relation between three different formulations of theorems on separable determination - one using the concept of rich families, second via the concept of suitable models and third, a new one, suggested in this paper, using the notion of $ω$-monotone mappings. In particular, we show that in Banach spaces all those formulations are in a sense equivalent and we give a positive answer to two questions of O. Kalenda and the author. Our results enable us to obtain new statements concerning separable determination of $σ$-porosity (and of similar notions) in the language of rich families; thus, not using any terminology from logic or set theory. Moreover, we prove that in Asplund spaces, generalized lushness is separably determined.
We prove that every Asplund lush space is generalized-lush using the method of separable reduction. This gives a partial positive answer to a question by Jan-David Hardtke.
We show a way of constructing projectional skeletons using the concept of rich families in Banach spaces which admit a projectional generator. Our next result is that a Banach space $X$ is Asplund and weakly compactly generated if and only if there exists a commutative 1-projectional skeleton $(Q_γ:\ γ\inΓ)$ on $X$ such that $(Q_γ^*:\ γ\inΓ)$ is a commutative 1-projectional skeleton on $X^*$. We consider both, real and also complex, Banach spaces.
In the framework of Asplund spaces, we use two equivalent instruments - rich families and suitable models from logic - for performing separable reductions of various statements on Frechet subdifferentiability of functions. This way, isometrical results are actually obtained and several existed proofs are substantially simplified. Everything is based on a new structural characterization of Asplund spaces.
We prove that a compact space is monotonically Sokolov if and only if it is monotonically $ω$-monolithic. This gives answers to several questions of R. Rojas-Hernandez and V. V. Tkachuk.
We compare two methods of proving separable reduction theorems in functional analysis -- the method of rich families and the method of elementary submodels. We show that any result proved using rich families holds also when formulated with elementary submodels and the converse is true in spaces with fundamental minimal system an in spaces of density $\aleph_1$. We do not know whether the converse is true in general. We apply our results to show that a projectional skeleton may be without loss of generality indexed by ranges of its projections.
We use a method involving elementary submodels and a partial converse of Foran lemma to prove separable reduction theorems concerning Suslin sigma-P-porous sets where "P" can be from a rather wide class of porosity-like relations in complete metric spaces. In particular, we separably reduce the notion of Suslin cone small set in Asplund spaces. As an application we prove a theorem stating that a continuous approximately convex function on an Asplund space is Frechet differentiable up to a cone small set.
We prove the existence of a simultaneous projectional skeleton for certain subspaces of $\mathcal{C}(K)$ spaces. This generalizes a result on simultaneous projectional resolutions of identity proved by M. Valdivia. We collect some consequences of this result. In particular we give a new characterization of Asplund spaces using the notion of projectional skeleton.
G. Godefroy and the second author of this note proved in 1988 that in duals to Asplund spaces there always exists a projectional resolution of the identity. A few years later, Ch. Stegall succeeded to drop from the original proof a deep lemma of S. Simons. Here, we rewrite the condensed argument of Ch. Stegall in a more transparent and detailed way. We actually show that this technology of Ch. Stegall leads to a bit stronger/richer object ---the so called projectional skeleton--- recently constructed by W. Kubiś, via S. Simons' lemma and with help of elementary submodels from logic.
We prove some generalizations of results concerning Valdivia compact spaces (equivalently spaces with a commutative retractional skeleton) to the spaces with a retractional skeleton (not necessarily commutative). Namely, we show that the dual unit ball of a Banach space is Corson provided the dual unit ball of every equivalent norm has a retractional skeleton. Another result to be mentioned is the following. Having a compact space K, we show that K is Corson if and only if every continuous image of K has a retractional skeleton.