Joanna Garbulińska-Wȩgrzyn

A metric space $X$ is {\em injective} if every non-expanding map $f \colon B\to X$ defined on a subspace $B$ of a metric space $A$ can be extended to a non-expanding map $\overline f \colon A\to X$. We prove that a metric space $X$ is a Lipschitz image of an injective metric space if and only if $X$ is {\em Lipschitz connected} in the sense that for every points $x,y\in X$, there exists a Lipschitz map $f \colon [0,1]\to X$ such that $f(0)=x$ and $f(1)=y$. In this case the metric space $X$ carries a well-defined intrinsic metric. A metric space $X$ is a Lipschitz image of a compact injective metric space if and only if $X$ is compact, Lipschitz connected and its intrinsic metric is totally bounded. A metric space $X$ is a Lipschitz image of a separable injective metric space if and only if $X$ is a Lipschitz image of the Urysohn universal metric space if and only if $X$ is analytic, Lipschitz connected and its intrinsic metric is separable.

Answering a question of Garbulińska-Wȩgrzyn and Kubiś, we prove that Gurariĭ operators form a dense $$G_\delta $$G δ -set in the space $$\mathcal B(\mathbb G)$$B ( G ) of all nonexpansive operators on the Gurariĭ space $$\mathbb G$$G , endowed with the strong operator topology. This implies that universal operators on $$\mathbb G$$G form a residual set in $$\mathcal B(\mathbb G)$$B ( G ) .

For a metric space $X$, let $\mathsf FX$ be the space of all non-empty finite subsets of $X$ endowed with the largest metric $d^1_{\mathsf FX}$ such that for every $n\in \mathbb N $ the map $X^n\to \mathsf FX$, $(x_1,\ldots ,x_n)\mapsto \{x_1,\ldots

We compare two types of universal operators constructed relatively recently by Cabello Sánchez, and the authors. The first operator $${\Omega }$$Ω acts on the Gurariĭ space, while the second one $${\mathbf {P}}_{{{\mathbb {S}}}}$$P S has values in a fixed separable Banach space $${{\mathbb {S}}}$$S . We show that if $${{\mathbb {S}}}$$S is the Gurariĭ space, then both operators are isometric. We also prove that, for a fixed space $${{\mathbb {S}}}$$S , the operator $${\mathbf {P}}_{{{\mathbb {S}}}}$$P S is isometrically unique. Finally, we show that $${\Omega }$$Ω is generic in the sense of a natural infinite game.

For a metric space $X$, let $\mathsf FX$ be the space of all nonempty finite subsets of $X$ endowed with the largest metric $d^1_{\mathsf FX}$ such that for every $n\in\mathbb N$ the map $X^n\to\mathsf FX$, $(x_1,\dots,x_n)\mapsto \{x_1,\dots,x_n\}$, is non-expanding with respect to the $\ell^1$-metric on $X^n$. We study the completion of the metric space $\mathsf F^1\!X=(\mathsf FX,d^1_{\mathsf FX})$ and prove that it coincides with the space $\mathsf Z^1\!X$ of nonempty compact subsets of $X$ that have zero length (defined with the help of graphs). We prove that each subset of zero length in a metric space has 1-dimensional Hausdorff measure zero. A subset $A$ of the real line has zero length if and only if its closure is compact and has Lebesgue measure zero. On the other hand, for every $n\ge 2$ the Euclidean space $\mathbb R^n$ contains a compact subset of 1-dimensional Hausdorff measure zero that fails to have zero length.

Let $${\mathcal {B}}$$B be a class of finite-dimensional Banach spaces. A $${\mathcal {B}}$$B -decomposed Banach spaceis a Banach spaceXendowed with a family $${\mathcal {B}}_X\subset {\mathcal {B}}$$BX⊂B of subspaces ofXsuch that each $$x\in X$$x∈X can be uniquely written as the sum of an unconditionally convergent series $$\sum _{B\in {\mathcal {B}}_X}x_B$$∑B∈BXxB for some $$(x_B)_{B\in {\mathcal {B}}_X}\in \prod _{B\in {\mathcal {B}}_X}B$$(xB)B∈BX∈∏B∈BXB . For every $$B\in {\mathcal {B}}_X$$B∈BX let $$\mathrm {pr}_B:X\rightarrow B$$prB:X→B denote the coordinate projection. Let $$C\subset [-1,1]$$C⊂[-1,1] be a closed convex set with $$C\cdot C\subset C$$C·C⊂C . TheC-decomposition constant $$K_C$$KC of a $${\mathcal {B}}$$B -decomposed Banach space $$(X,{\mathcal {B}}_X)$$(X,BX) is the smallest number $$K_C$$KC such that for every function $$\alpha :{\mathcal {F}}\rightarrow C$$α:F→C from a finite subset $${\mathcal {F}}\subset {\mathcal {B}}_X$$F⊂BX the operator $$T_\alpha =\sum _{B\in {\mathcal {F}}}\alpha (B)\cdot \mathrm {pr}_B$$Tα=∑B∈Fα(B)·prB has norm $$\Vert T_\alpha \Vert \le K_C$$‖Tα‖≤KC . By $$\varvec{{\mathcal {B}}}_C$$BC we denote the class of $${\mathcal {B}}$$B -decomposed Banach spaces withC-decomposition constant $$K_C\le 1$$KC≤1 . Using the technique of Fraïssé theory, we construct a rational $${\mathcal {B}}$$B -decomposed Banach space $$\mathbb {U}_C\in \varvec{{\mathcal {B}}}_C$$UC∈BC which contains an almost isometric copy of each $${\mathcal {B}}$$B -decomposed Banach space $$X\in \varvec{{\mathcal {B}}}_C$$X∈BC . If $${\mathcal {B}}$$B is the class of all 1-dimensional (resp. finite-dimensional) Banach spaces, then $$\mathbb {U}_{C}$$UC is isomorphic to the complementably universal Banach space for the class of Banach spaces with an unconditional (f.d.) basis, constructed by Pełczyński (and Wojtaszczyk).

For a constant $K\geq 1$ let $\mathfrak{B}_K$ be the class of pairs $(X,(\mathbf e_n)_{n\in\omega})$ consisting of a Banach space $X$ and an unconditional Schauder basis $(\mathbf e_n)_{n\in\omega}$ for $X$, having the unconditional basic constant $K_u\leq K$. Such pairs are called $K$-based Banach spaces. A based Banach space $X$ is rational if the unit ball of any finite-dimensional subspace spanned by finitely many basic vectors is a polyhedron whose vertices have rational coordinates in the Schauder basis of $X$. Using the technique of Fra\"iss\'e theory, we construct a rational $K$-based Banach space $\big(\mathbb U_K,(\mathbf e_n)_{n\in\omega}\big)$ which is $\mathfrak{RI}_K$-universal in the sense that each basis preserving isometry $f:\Lambda\to\mathbb U_K$ defined on a based subspace $\Lambda$ of a finite-dimensional rational $K$-based Banach space $A$ extends to a basis preserving isometry $\bar f:A\to\mathbb U_K$ of the based Banach space $A$. We also prove that the $K$-based Banach space $\mathbb U_K$ is almost $\mathfrak{FI}_1$-universal in the sense that any base preserving $\varepsilon$-isometry $f:\Lambda\to\mathbb U_K$ defined on a based subspace $\Lambda$ of a finite-dimensional $1$-based Banach space $A$ extends to a base preserving $\varepsilon$-isometry $\bar f:A\to\mathbb U_K$ of the based Banach space $A$. On the other hand, we show that no almost $\mathfrak{FI}_K$-universal based Banach space exists for $K>1$. The Banach space $\mathbb U_K$ is isomorphic to the complementably universal Banach space for the class of Banach spaces with an unconditional Schauder basis, constructed by Pe\l czy\'nski in 1969.

Using the technique of Fra\"iss\'e theory, for every constant $K\ge 1$ we consruct a universal object in the class of Banach spaces with normalized $K$-suppression unconditional Schauder bases.

Using the technique of Fraïssé theory, for every constant $K\ge 1$, we construct a universal object $\mathbb U_K$ in the class of Banach spaces possessing a normalized $K$-suppression unconditional Schauder basis.