arXiv subjects

Michal Doucha

Publications and source records attributed to Michal Doucha.

At least 37 records · Page 2Linked to original sources

arXiv2020

Lipschitz algebras and Lipschitz-free spaces over unbounded 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.

math.FA↗

arXiv2020

On Dual surjunctivity and applications

We explore the dual version of Gottschalk's conjecture recently introduced by Capobianco, Kari, and Taati, and the notion of dual surjunctivity in general. We show that dual surjunctive groups satisfy Kaplansky's direct finiteness conjecture for all fields of positive characteristic. By quantifying the notions of injectivity and post-surjectivity for cellular automata, we show that the image of the full topological shift under an injective cellular automaton is a subshift of finite type in a quantitative way. Moreover we show that dual surjunctive groups are closed under ultraproducts, under elementary equivalence, and under certain semidirect products (using the ideas of Arzhantseva and Gal for the latter); they form a closed subset in the space of marked groups, fully residually dual surjunctive groups are dual surjunctive, etc. We also consider dual surjunctive systems for more general dynamical systems, namely for certain expansive algebraic actions, employing results of Chung and Li.

math.GR↗

arXiv2020

Lipschitz free spaces isomorphic to their infinite sums and geometric applications

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 $p=1$.

math.FA↗

arXiv2020

Structure of the Lipschitz free $p$-spaces $\mathcal{F}_p(\mathbb{Z}^d)$ and $\mathcal{F}_p(\mathbb{R}^d)$ for $0<p\le 1$

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 $0<p<1$.

math.FA↗

arXiv2020

Complexity of distances: Theory of generalized analytic equivalence relations

We generalize the notion of analytic/Borel equivalence relations, orbit equivalence relations, and Borel reductions between them to their continuous and quantitative counterparts: analytic/Borel pseudometrics, orbit pseudometrics, and Borel reductions between them. We motivate these concepts on examples and we set some basic general theory. We illustrate the new notion of reduction by showing that the Gromov-Hausdorff distance maintains the same complexity if it is defined on the class of all Polish metric spaces, spaces bounded from below, from above, and from both below and above. Then we show that $E_1$ is not reducible to equivalences induced by orbit pseudometrics, generalizing the seminal result of Kechris and Louveau. We answer in negative a question of Ben-Yaacov, Doucha, Nies, and Tsankov on whether balls in the Gromov-Hausdorff and Kadets distances are Borel. In appendix, we provide new methods using games showing that the distance-zero classes in certain pseudometrics are Borel, extending the results of Ben Yaacov, Doucha, Nies, and Tsankov. There is a complementary paper of the authors where reductions between the most common pseudometrics from functional analysis and metric geometry are provided.

math.FA↗

arXiv2019

Embeddability of $\ell_{p}$ and bases in Lipschitz free $p$-spaces for $0<p\leq 1$

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.

math.FA↗

arXiv2019

Generic representations of countable groups

The paper is devoted to a study of generic representations (homomorphisms) of discrete countable groups $Γ$ in Polish groups $G$, i.e. those elements in the Polish space $\mathrm{Rep}(Γ,G)$ of all representations of $Γ$ in $G$, whose orbit under the conjugation action of $G$ on $\mathrm{Rep}(Γ,G)$ is comeager. We investigate a closely related notion of finite approximability of actions on countable structures such as tournaments or $K_n$-free graphs, and we show its connections with Ribes-Zalesski-like properties of the acting groups. We prove that $\mathbb{N}$ has a generic representation in the automorphism group of the random tournament (i.e., there is a comeager conjugacy class in this group). We formulate a Ribes-Zalesskii-like condition on a group that guarantees finite approximability of its actions on tournaments. We also provide a simpler proof of a result of Glasner, Kitroser and Melleray characterizing groups with a generic permutation representation. We also investigate representations of infinite groups $Γ$ in automorphism groups of metric structures such as the isometry group $\mathrm{Iso}(\mathbb{U})$ of the Urysohn space, isometry group $\mathrm{Iso}(\mathbb{U}_1)$ of the Urysohn sphere, or the linear isometry group $\mbox{LIso}(\mathbb{G})$ of the Gurarii space. We show that the conjugation action of $\mathrm{Iso}(\mathbb{U})$ on $\mathrm{Rep}(Γ,\mathrm{Iso}(\mathbb{U}))$ is generically turbulent answering a question of Kechris and Rosendal.

math.GR↗

arXiv2019

Lipschitz free $p$-spaces for $0<p<1$

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.

math.FA↗

arXiv2019

Generic norms and metrics on countable abelian groups

For a countable abelian group $G$ we investigate generic properties of the space of all invariant metrics on $G$. We prove that for every such an unbounded group $G$, i.e. group which has elements of arbitrarily high order, there is a dense set of invariant metrics on $G$ which make $G$ isometric to the rational Urysohn space, and a comeager set of invariant metrics such that the completion is isometric to the Urysohn space. This generalizes results of Cameron and Vershik, Niemiec, and the author. Then we prove that for every $G$ such that $G\cong \bigoplus_\mathbb{N} G$ there is a comeager set of invariant metrics on $G$ such that all of them give rise to the same metric group after completion. If moreover $G$ is unbounded, then using a result of Melleray and Tsankov we get that the completion is extremely amenable.

math.GN↗

arXiv2019

Isomorphisms between spaces of Lipschitz functions

We develop tools for proving isomorphisms of normed spaces of Lipschitz functions over various doubling metric spaces and Banach spaces. In particular, we show that $\operatorname{Lip}_0(\mathbb{Z}^d)\simeq\operatorname{Lip}_0(\mathbb{R}^d)$, for all $d\in\mathbb{N}$. More generally, we e.g. show that $\operatorname{Lip}_0(Γ)\simeq \operatorname{Lip}_0(G)$, where $Γ$ is from a large class of finitely generated nilpotent groups and $G$ is its Mal'cev closure; or that $\operatorname{Lip}_0(\ell_p)\simeq\operatorname{Lip}_0(L_p)$, for all $1\leq p<\infty$. We leave a large area for further possible research.

math.FA↗

arXiv2018

Universal actions and representations of locally finite groups on metric spaces

We construct a universal action of a countable locally finite group (the Hall's group) on a separable metric space by isometries. This single action contains all actions of all countable locally finite groups on all separable metric spaces as subactions. The main ingredient is the amalgamation of actions by isometries. We show that an equivalence class of this universal action is generic. We show that the restriction to locally finite groups in our results is necessary as analogous results do not hold for infinite non-locally finite groups. We discuss the problem also for actions by linear isometries on Banach spaces.

math.GR↗

arXiv2018

Metrically universal abelian groups

We give a positive answer to the question of Shkarin (\emph{On universal abelian topological groups}, Mat. Sb. 190 (1999), no. 7, 127-144) whether there exists a metrically universal abelian separable group equipped with invariant metric. Our construction also gives an example of a group structure on the Urysohn universal space that is substantially different from the previously known examples. Under some cardinal arithmetic assumptions, our results generalize to higher cardinalities.

math.GN↗

arXiv2018

Scott rank of Polish metric spaces

Following the work of Friedman, Koerwien, Nies and Schlicht we positively answer their question whether the Scott rank of Polish metric spaces is countable.

math.LO↗

arXiv2018

Non-abelian group structure on the Urysohn space

Following the continuing interest in the Urysohn space and, more specifically, the recent problem area of finding and comparing group structures on the Urysohn space we prove that there exists a non-abelian group structure on the Urysohn universal metric space. More precisely, we introduce a variant of the Graev metric that enables us to construct a free group with countably many generators equipped with a two-sided invariant metric that is isometric to the rational Urysohn space. We provide several open questions and problems related to this subject.

math.GN↗

arXiv2018

Non-universality of automorphism groups of uncountable ultrahomogeneous structures

In \cite{MbPe}, Mbombo and Pestov prove that the group of isometries of the generalized Urysohn space of density $κ$, for uncountable $κ$ such that $κ^{<κ}=κ$, is not a universal topological group of weight $κ$. We investigate automorphism groups of other uncountable ultrahomogeneous structures and prove that they are rarely universal topological groups for the corresponding classes. Our list of uncountable ultrahomogeneous structures includes random uncountable graph, tournament, linear order, partial order, group. That is in contrast with similar results obtained for automorphism groups of countable (separable) ultrahomogeneous structures. We also provide a more direct and shorter proof of the Mbombo-Pestov's result.

math.LO↗

arXiv2018

Metrical universality for groups

We prove that for any constant $K>0$ there exists a separable group equipped with a complete bi-invariant metric bounded by $K$, isometric to the Urysohn sphere of diameter $K$, that is of `almost-universal disposition'. It is thus an object in the category of separable groups with bi-invariant metric analogous in its properties to the Gurarij space from the category of separable Banach spaces. We show that this group contains an isometric copy of any separable group equipped with bi-invariant metric bounded by $K$. As a consequence, we get that it is a universal Polish group admitting compatible bi-invariant metric, resp. universal second countable SIN group. Moreover, the almost-universal disposition shows that the automorphism group of this group is rich and it characterizes the group uniquely up to isometric isomorphism. We also show that this group is in a certain sense generic in the class of separable group with bi-invariant metric (bounded by $K$). On the other hand, we prove there is no metrically universal separable group with bi-invariant metric when there is no restriction on diameter. The same is true for separable locally compact groups with bi-invariant metric. Assuming the generalized continuum hypothesis, we prove that there exists a metrically universal (unbounded) group of density $κ$ with bi-invariant metric for any uncountable cardinal $κ$. We moreover deduce that under GCH there is a universal SIN group of weight $κ$ for any infinite cardinal $κ$.

math.GR↗

arXiv2018

Universal and homogeneous structures on the Urysohn and Gurarij spaces

Using Fra\" iss\' e theoretic methods we enrich the Urysohn universal space by universal and homogeneous closed relations, retractions, closed subsets of the product of the Urysohn space itself and some fixed compact metric space, $L$-Lipschitz map to a fixed Polish metric space. The latter lifts to a universal linear operator of norm $L$ on the Lispchitz-free space of the Urysohn space. Moreover, we enrich the Gurarij space by a universal and homogeneous closed subspace and norm one projection onto a $1$-complemented subspace. We construct the Gurarij space by the classical Fra\" iss\' e theoretic approach.

math.GN↗

arXiv2018

Lipschitz-free spaces over ultrametric spaces

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.

math.FA↗