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Michal Doucha

Publications and source records attributed to Michal Doucha.

At least 19 recordsLinked to original sources
arXiv2026

Invariant pointwise closed subspaces of Lipschitz spaces and their preduals

Motivated by both the research on Lipschitz-free spaces and on Lipschitz harmonic functions on graphs, we study invariant pointwise closed subspaces of spaces of Lipschitz functions over graphs, with special emphasis on finitely generated groups as graphs. Such spaces form a natural class of $\text{weak}^*$-closed subspaces, that can be fully described in some cases, and hence have canonical quotient preduals of the corresponding Lipschitz-free spaces. We show that these spaces are described by finite local constraints; in the group case as Lipschitz solutions of systems of convolution equations. We describe and characterize their preduals via a universal property and show that whenever they contain a non-zero element with a $c_0$-gradient, then they contain $\ell_\infty$, and consequently their preduals contain a complemented copy of $\ell_1$. A guiding question is whether this class of Lipschitz spaces and their preduals contains an infinite-dimensional reflexive Banach space. In this regard, the main result of the paper is the following dichotomy proved using abstract harmonic analysis. Every translation-invariant pointwise closed subspace of the Lipschitz space over $\mathbb{Z}^d$ is either finite-dimensional or it is non-separable --in particular, the corresponding predual is either finite-dimensional or non-reflexive.

math.FA↗
arXiv2025

Group equivariant Radon-Nikod\'ym property and its characterizations

We introduce and study equivariant versions of the Radon-Nikod\'ym property for Banach spaces, together with the closely related notions such as dentability, the Bishop-Phelps and Krein-Milman properties, and Lindenstrauss' property A, all considered in the presence of a continuous group action by linear isometries. While in the classical setting the Radon-Nikod\'ym property, the Bishop-Phelps property and dentability are equivalent, the equivariant situation turns out to depend essentially on the acting group and requires non-trivial tools from abstract harmonic analysis and representation theory. We establish several implications among the equivariant counterparts of these properties. Namely, given a compact group $G$, the $G$-Bishop-Phelps property implies strong $G$-dentability, which in turn implies the $G$-Krein-Milman property and the classical Bishop-Phelps property, for any $G$-Banach space. Moreover, given a locally compact and second countable group $G$, the $G$-Radon-Nikod\'ym property is equivalent to the classical Radon-Nikod\'ym property, for any $G$-Banach space.

math.FA↗
arXiv2025

Invariant strictly convex renormings

Motivated by the question of Mikael de la Salle, we investigate the problem of the existence of equivalent strictly convex norms on Banach spaces that are invariant with respect to an action of a group by linear isometries. We develop various tools for constructing such norms and prove several preservation results. We also answer positively a question of Antunes, Ferenczi, Grivaux and Rosendal whether there is a strictly convex renorming of $c$ invariant with respect to its full linear isometry group. Finally, we specialize to the spaces $L_1[0,1]$ and $C(K)$, where $K$ is compact Hausdorff, and indicate that amenability of groups plays a role in this problem.

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arXiv2025

Dense and comeager conjugacy classes in zero-dimensional dynamics

Given a countable group $G$, we initiate a systematic study of the Polish spaces of all minimal and topologically transitive actions of $G$ on the Cantor space by homeomorphisms, with a focus on the existence of comeager conjugacy classes in these spaces. We develop a general model-theoretic framework to study this and related questions, recovering on the way many existing results from the literature. A substantial part of the paper is devoted to actions of free groups. We show that in that case, there is a comeager conjugacy class in the space of minimal actions, as well as in the space of minimal, probability measure-preserving actions. The first one is the Fra\"iss\'e limit of all sofic minimal subshifts and the second, the universal profinite action. The case of the integers was already treated by Hochman and there the two actions coincide with the universal odometer. In the non-abelian case, they are substantially different and new techniques are required. In the opposite direction, if $G$ is an amenable group which is not finitely generated, we show that there is no comeager conjugacy class in the space of all actions, and if $G$ is locally finite, also in the space of minimal actions. Finally, we study the question of existence of a dense conjugacy class in the space of topologically transitive actions. We show that if $G$ is hyperbolic or virtually polycyclic, then such a dense conjugacy class exists iff $G$ is virtually cyclic, suggesting that the case of the integers may be exceptional.

math.DS↗
arXiv2025

Automorphism groups of measures on the Cantor space. Part I: Good measures and Rokhlin properties

We study criteria for the existence of a dense or comeager conjugacy class in the automorphism group of a given measure on the Cantor space. We concentrate on good measures, defined by Akin [\emph{Trans.\ Amer.\ Math.\ Soc.} \textbf{357} (2005), no. 7, 2681--2722], which we characterize as a particular subclass of ultrahomogeneous measures. We determine good measures with rational values on clopen sets whose automorphism group admits a comeager conjugacy class. Our approach uses the Fra\"{i}ss\'{e} theory.

math.LO↗
arXiv2024

Guarded Fra\"iss\'e Banach spaces

We characterize separable Banach spaces having $G_\delta$ isometry classes in the Polish codings $\mathcal{P}$, $\mathcal{P}_\infty$ and $\mathcal{B}$ introduced by C\'uth-Dole\v{z}al-Doucha-Kurka [13] as those being guarded Fra\"iss\'e, a weakening of the notion of Fra\"iss\'e Banach spaces defined by Ferenczi-Lopez-Abad-Mbombo-Todorcevic [18]. We prove a Fra\"iss\'e correspondence for those spaces and make links with the notion of $\omega$-categoricity from continuous logic, showing that $\omega$-categorical Banach spaces are a natural source of guarded Fra\"iss\'e Banach spaces. Using those results, we prove that for many values of $(p, q)$, the Banach space $L_p(L_q)$ has a $G_\delta$ isometry class; we precisely characterize those values.

math.FA↗
arXiv2024

Isometries of Lipschitz-free Banach spaces

We describe surjective linear isometries and linear isometry groups of a large class of Lipschitz-free spaces that includes e.g. Lipschitz-free spaces over any graph. We define the notion of a Lipschitz-free rigid metric space whose Lipschitz-free space only admits surjective linear isometries coming from surjective dilations (i.e. rescaled isometries) of the metric space itself. We show this class of metric spaces is surprisingly rich and contains all $3$-connected graphs as well as geometric examples such as non-abelian Carnot groups with horizontally strictly convex norms. We prove that every metric space isometrically embeds into a Lipschitz-free rigid space that has only three more points.

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arXiv2023

Lie theoretic approach to unitary groups of $C^*$-algebras

Following Robert's [26], we study the structure of unitary groups and groups of approximately inner automorphisms of unital $C^*$-algebras, taking advantage of the former being Banach-Lie groups. For a given unital $C^*$-algebra $A$, we provide a description of the closed normal subgroup structure of the connected component of the identity of the unitary group, denoted by $U_A$, resp. of the subgroup of approximately inner automorphisms induced by the connected component of the identity of the unitary group, denoted by $V_A$, in terms of perfect ideals, i.e. ideals admitting no characters. When the unital algebra is locally AF, we show that there is a one-to-one correspondence between closed normal subgroups of $V_A$ and perfect ideals of the algebra, which can be in the separable case conveniently described using Bratteli diagrams; in particular showing that every closed normal subgroup of $V_A$ is perfect. We also characterize unital $C^*$-algebras $A$ such that $U_A$, resp. $V_A$ are topologically simple, generalizing the main results from [26]. In the other way round, under certain conditions, we characterize simplicity of the algebra in terms of the structure of the unitary group. This in particular applies to reduced group $C^*$-algebras of discrete groups and we show that when $A$ is a reduced group $C^*$-algebra of a non-amenable countable discrete group, then $A$ is simple if and only if $U_A/\mathbb{T}$ is topologically simple.

math.OA↗
arXiv2023

An application of Kirchberg's lemma on central sequence algebras to groups of approximately inner automorphisms

We revisit a well-known "surjectivity onto quotient" type lemma of Kirchberg on the central sequence algebra of a separable unital ${\rm C}^*$-algebra, and use it to prove a "surjectivity onto quotient" result on approximately inner automorphisms of a separable unital ${\rm C}^*$-algebra of stable rank one, which we can partially upgrade also to the non-separable case.

math.OA↗
arXiv2022

Strong topological Rokhlin property, shadowing, and symbolic dynamics of countable groups

A countable group $G$ has the strong topological Rokhlin property (STRP) if it admits a continuous action on the Cantor space with a comeager conjugacy class. We show that having the STRP is a symbolic dynamical property. We prove that a countable group $G$ has the STRP if and only if certain sofic subshifts over $G$ are dense in the space of subshifts. A sufficient condition is that isolated shifts over $G$ are dense in the space of all subshifts. We provide numerous applications including the proof that a group that decomposes as a free product of finite or cyclic groups has the STRP. We show that finitely generated nilpotent groups do not have the STRP unless they are virtually cyclic; the same is true for many groups of the form $G_1\times G_2\times G_3$ where each factor is recursively presented. We show that a large class of non-finitely generated groups do not have the STRP, this includes any group with infinitely generated center and the Hall universal locally finite group. We find a very strong connection between the STRP and shadowing, a.k.a. pseudo-orbit tracing property. We show that shadowing is generic for actions of a finitely generated group $G$ if and only if $G$ has the STRP.

math.DS↗
arXiv2022

Polish spaces of Banach spaces. Complexity of isometry and isomorphism classes

We study the complexities of isometry and isomorphism classes of separable Banach spaces in the Polish spaces of Banach spaces recently introduced and investigated by the authors in [14]. We obtain sharp results concerning the most classical separable Banach spaces. We prove that the infinite-dimensional separable Hilbert space is characterized as the unique separable infinite-dimensional Banach space whose isometry class is closed, and also as the unique separable infinite-dimensional Banach space whose isomorphism class is $F_\sigma$. For $p\in\left[1,2\right)\cup\left(2,\infty\right)$, we show that the isometry classes of $L_p[0,1]$ and $\ell_p$ are $G_\delta$-complete sets and $F_{\sigma\delta}$-complete sets, respectively. Then we show that the isometry class of $c_0$ is an $F_{\sigma\delta}$-complete set. Additionally, we compute the complexities of many other natural classes of separable Banach spaces; for instance, the class of separable $\mathcal{L}_{p,\lambda+}$-spaces, for $p,\lambda\geq 1$, is shown to be a $G_\delta$-set, the class of superreflexive spaces is shown to be an $F_{\sigma\delta}$-set, and the class of spaces with local $\Pi$-basis structure is shown to be a $\boldsymbol{\Sigma}^0_6$-set. The paper is concluded with many open problems and suggestions for a future research.

math.FA↗
arXiv2021

Garden of Eden and weakly periodic points for certain expansive actions of groups

We present several applications of the weak specification property and certain topological Markov properties, recently introduced by S. Barbieri, F. García-Ramos and H. Li, and implied by the pseudo-orbit tracing property, for general expansive group actions on compact spaces. First we show that any expansive action of a countable amenable group on a compact metrizable space satisfying the weak specification and strong topological Markov properties satisfies the Moore property, i.e. every surjective automorphism of such dynamical system is pre-injective. This together with an earlier result of H. Li (where the strong topological Markov property is not needed) of the Myhill property, which we also re-prove here, establishes the Garden of Eden theorem for all expansive actions of countable amenable groups on compact metrizable spaces satisfying the weak specification and strong topological Markov properties. We hint how to easily generalize this result even for uncountable amenable groups and general compact, not necessarily metrizable, spaces. Second, we generalize the recent result of D. B. Cohen that any subshift of finite type of a finitely generated group having at least two ends has weakly periodic points. We show that every expansive action of such a group having a certain Markov topological property, again implied by the pseudo-orbit tracing property, has a weakly periodic point. If it has additionally the weak specification property, the set of such points is dense.

math.DS↗
arXiv2021

Projections in Lipschitz-free spaces induced by group actions

We show that given a compact group $G$ acting continuously on a metric space $M$ by bi-Lipschitz bijections with uniformly bounded norms, the Lipschitz-free space over the space of orbits $M/G$ (endowed with Hausdorff distance) is complemented in the Lipschitz-free space over $M$. We also investigate the more general case when $G$ is amenable, locally compact or SIN and its action has bounded orbits. Then we get that the space of Lipschitz functions $Lip_0(M/G)$ is complemented in $Lip_0(M)$. Moreover, if the Lipschitz-free space over $M$, $F(M)$, is complemented in its bidual, several sufficient conditions on when $F(M/G)$ is complemented in $F(M)$ are given. Some applications are discussed. The paper contains preliminaries on projections induced by actions of amenable groups on general Banach spaces.

math.FA↗
arXiv2020

Large scale geometry of Banach-Lie groups

We initiate the large scale geometric study of Banach-Lie groups, especially of linear Banach-Lie groups. We show that the exponential length, originally introduced by Ringrose for unitary groups of $C^*$-algebras, defines the quasi-isometry type of any connected Banach-Lie group. As an illustrative example, we consider unitary groups of separable abelian unital $C^*$-algebras with spectrum having finitely many components, which we classify up to topological isomorphism and up to quasi-isometry, in order to highlight the difference. The main results then concern the Haagerup property, and Properties (T) and (FH). We present the first non-trivial non-abelian and non-localy compact groups having the Haagerup property, most of them being non-amenable. These are the groups $\mathcal{U}_2(M,τ)$, where $M$ is a semifinite von Neumann algebra with a normal faithful semifinite trace $τ$. Finally, we investigate the groups $\mathrm{E}_n(A)$, which are closed subgroups of $\mathrm{GL}(n,A)$ generated by elementary matrices, where $A$ is a unital Banach algebra. We show that for $n\geq 3$, all these groups have Property (T) and they are unbounded, so they have Property (FH) non-trivially. On the other hand, if $A$ is an infinite-dimensional unital $C^*$-algebra, then $\mathrm{E}_2(A)$ does not have the Haagerup property. If $A$ is moreover abelian and separable, then $\mathrm{SL}(2,A)$ does not have the Haagerup property.

math.OA↗
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 $p=1$.

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