Plasticity of the unit sphere of the space $c_0$

Introduction

Definition. A metric space $(M,d)$ is called plastic if every bijective non-expansive map $F:M\to M$ is an isometry.

Recall that a map $F$ is called non-expansive (non-contractive) if for any $x,y\in M$

A map $F$ is called non-contractive if for any $x,y\in M$ the inverse inequality holds. In [2], Cascales, Kadets, Orihuela, and Wingler posed the following question:

This question has a number of partial positive results (see, for example, [2, 3, 5, 6, 13] and references therein), but is still open in full generality in infinite dimensions. In finite dimensions, the answer is positive due to the result on totally bounded metric spaces obtained by Naimpally, Piotrowski, and Wingler [10]. The problem of plasticity of the unit balls of Banach spaces may be considered as a continuation of the well-known results of Mazur–Ulam [9] extended by Mankiewicz [8].

In 1987, Tingley [12] raised the question of whether the unit balls in the aforementioned theorem could be replaced by unit spheres:

Despite numerous partial results for specific spaces (see, for example, [4], [11]), Tingley’s problem remains a challenging open question in dimensions three and higher. Moreover, it has been fully resolved in the two-dimensional setting by Banakh only in 2022 [1]. As its natural continuation, one may consider the plasticity problem for the unit spheres of Banach spaces, posed by V. Kadets in personal communication, 2024.

In [7] this question was answered positively for Hilbert spaces, $\ell_1$, $\ell_\infty$, and $c$. Observe also, for the unit ball of the space $c_0$ only conditional plasticity was shown. Namely, a bijective non-expansive map $F:B_{c_0}\to B_{c_0}$ is an isometry if $F$ is continuous [6]. It is currently unknown whether this additional condition may be omitted. As the title says, the main objective of this paper is to prove the plasticity (without any additional conditions) of $S_{c_0}$.

To facilitate our analysis, we introduce the necessary notation and structural definitions used throughout the text. For a real Banach space $X$, let $S_X$ and $B_X$ denote its unit sphere and closed unit

2020 Mathematics Subject Classification. 46B20, 46C05.

Key words and phrases. Non-expansive map, non-contractive map, unit sphere, isometry, expand-contract plastic space.

The author was supported by the National Research Foundation of Ukraine, project No. 2025.07/0369 “Qualitative methods of nonlinear analysis of heterogeneous structures”. ball, respectively. In the context of the classical sequence space $c_0$, an element $x \in S_{c_0}$ is represented as a sequence $x = (x^1, x^2, \ldots)$ such that $\lim_{n \to\infty} x^n = 0$ and $\max_{n \in\mathbb{N}} |x^n| = 1$.

We denote by $S^*_{c_0}$ the dense subset of $S_{c_0}$ consisting of all elements with finite support. For any vector $x \in c_0$, its support is defined as the set of indices corresponding to non-zero coordinates:

Thus, $x \in S^*_{c_0}$ if and only if $|\operatorname{supp}(x)| < \infty$.

Furthermore, let $\{e_n\}_{n \in\mathbb{N}}$ be the standard unit vector basis of $c_0$, where $e_n$ has 1 at the $n$-th position and zeros elsewhere. We define $E$ as the set consisting of all standard basis vectors and their negatives:

Evidently, $E \subset S^*_{c_0} \subset S_{c_0}$. For any real number $t \in\mathbb{R}$, its signum function is denoted by $\operatorname{sgn}(t)$.

The paper is organized as follows. In the second section, we collect and modify some results known for the unit spheres of some other sequence spaces or for $B_{c_0}$ to adapt them to the case of $S_{c_0}$. These results imply that any bijective non-expansive mapping is tightly bound to a coordinate permutation and a sign-switching sequence. Roughly speaking, we “turn off” this coordinate permutation and sign-switching and obtain a function $\widetilde{F}$. This section also includes some previously known or slightly modified properties of this function. Section 3 is devoted to further investigation of the properties of $\widetilde{F}$, which finally allowed us to see that $\widetilde{F}$ reduces to identity, which in turn means that $F$ reduces to an isometry.

Preliminaries

The analog of the first lemma of this section is Lemma 3.1 from [6], which was proved for $B_{c_0}$ and $B_c$. The “only if” part was also proved in Lemma 4.1 in [7] for the unit spheres of $\ell_\infty$ and $c$. Although this result cannot be considered as a completely new one, for the reader’s convenience, we give it with proof.

Proof. Let us first show the “if” part. Suppose that without loss of generality $p=e_n$, $q=-e_n$ and fix arbitrary $x \in S_{c_0}$. If $x^n=0$, then both inequalities from the lemma statement hold. If $x^n>0$, then $\lVert x-p\rVert\le1$, otherwise $\lVert x-q\rVert\le1$.

Conversely, let us prove the “only if” part. Suppose by contradiction that $|\operatorname{supp}(p)|>1$. Then there exist two distinct indices $k$ and $j$ such that $p^k \ne0$ and $p^j \ne0$ and an index $l$, which without loss of generality we may assume different from $k$, such that $q^l \ne0$. Let us construct a vector $x \in S_{c_0}$ as follows: set $x^k=-\operatorname{sgn}(p^k)$, and $x^l=-\operatorname{sgn}(q^l)$. All other coordinates of $x$ are set to zero. Then we clearly obtain:

and by the choice of $x^l$, we also have $\lVert x-q\rVert\ge|x^l-q^l|>1$, which contradicts the assumption.

Thus, $|\operatorname{supp}(p)|=|\operatorname{supp}(q)|=1$. Since $p,q \in S_{c_0}$, this implies $p=\pm e_n$ for some $n \in\mathbb{N}$, meaning $p \in E$. By symmetry, we also get $q \in E$, so $q=\pm e_m$. If $m \ne n$, we can easily find $x \in S_{c_0}$ to violate the condition. Finally, if $q=p$, then choosing $x=-p$ yields $\lVert x-p\rVert=\lVert-2p\rVert=2>1$, which is another contradiction. Therefore, we have $q=-p$. □

The next lemma was also proved in [7] (Lemma 4.2) for $c$ and $\ell_\infty$, however, it also works for $c_0$.

Proof. For every $y \in S_{c_0}$ either $\lVert y-x\rVert\le1$ or $\lVert y+x\rVert\le1$, and the non-expansiveness of $F$ implies that $\min(\lVert F(y)-F(x)\rVert,\lVert F(y)-F(-x)\rVert)\le1$ for every $y \in S_{c_0}$. $F$ is bijective, so $\min(\lVert y-F(x)\rVert,\lVert y-F(-x)\rVert)\le1$ for every $y \in S_{c_0}$. Now all that remains is to apply the previous lemma. □

The next lemma was extracted from Lemma 3.2 in [6], where the same was proved for $B_{c_0}$.

Proof. Let us fix an arbitrary $n\in\mathbb{N}$. By Lemma 2.2, for the elements $e_n$ and $-e_n$, we have $F(e_n)=e_k$ or $F(e_n)=-e_k$, and correspondingly, $F(-e_n)=-e_k$ or $F(-e_n)=e_k$ for some $k\in\mathbb{N}$. Define $\sigma(n)=k$ and $\alpha^n=(F(e_n))^k$. The mapping $\sigma$ defined in this way is injective due to the injectivity of $F$.

In what follows, we show that conditions 1)–3) hold for all $x\in S_{c_0}$ and $n\in\mathbb{N}$. Since $S^*_{c_0}$ is dense in $S_{c_0}$ and $F$ is continuous, it suffices to verify these conditions only for $x\in S^*_{c_0}$, that is, for $x\in S_{c_0}$ with $|\operatorname{supp}(x)|<\infty$.

For every $n\in\operatorname{supp}(x)$ define $l_n=-\operatorname{sgn}(x^n)e_n$. Observe that for every $t\in S_{c_0}$, either $\lVert t-x\rVert\leq1$ or there exists $n\in\operatorname{supp}(x)$ such that $\lVert t-l_n\rVert\leq1$. To demonstrate this, let us fix $t\in S_{c_0}$ and suppose $\lVert t-l_n\rVert>1$ for every $n\in\operatorname{supp}(x)$. By the definition of $l_n$, this means $\operatorname{sgn}(x^nt^n)>0$ for every $n\in\operatorname{supp}(x)$ and thus, $\lVert t-x\rVert\leq1$. Since $F$ is non-expansive and bijective, for every $t\in S_{c_0}$, either $\lVert t-F(x)\rVert\leq1$ or there is $n\in\operatorname{supp}(x)$ such that $\lVert t-F(l_n)\rVert\leq1$. Further, we will construct a special sequence $z\in S_{c_0}$ and show $\lVert z-F(x)\rVert\leq1$. By the construction of $z$, the latter inequality will automatically imply items (2) and (3). Let

Observe by its definition, $z\in S_{c_0}$ and $\lVert z-F(l_n)\rVert>1$ for every $n\in\operatorname{supp}(x)$, so $\lVert z-F(x)\rVert\leq1$, and thus items (2) and (3) are proved.

It remains to prove item (1). Let us fix $k\notin\sigma(\operatorname{supp}(x))$ and change the definition of $z$ such that $z^k=1$. We still have $z\in S^*_{c_0}$ and $\lVert z-F(l_n)\rVert>1$ for every $n\in\operatorname{supp}(x)$. Thus, $\lVert z-F(x)\rVert\leq1$, which implies $F(x)^k\geq0$. On the other hand, setting $z^k=-1$, we have the opposite inequality $F(x)^k\leq0$. Thus, item (1) is proved.

The last step is to prove surjectivity of $\sigma$. Let us argue by contradiction. Suppose there is $j\notin\sigma(\mathbb{N})$. From the above we have $F(x)^j=0$ for every $x\in S^*_{c_0}$. Using the continuity of $F$ and density of $S^*_{c_0}$ in $S_{c_0}$ once again, we obtain $F(x)^j=0$ for every $x\in S_{c_0}$. This contradiction to the surjectivity of $F$ completes the proof.

:::

The preceding lemma yields a bijection $\sigma:\mathbb{N}\to\mathbb{N}$ and a sequence $\alpha:\mathbb{N}\to\{-1,1\}$. Let us define the following isometric automorphism $A$ of $c_0$ by

for all $n\in\mathbb{N}$. Denote the restriction of this automorphism to $S_{c_0}$ by $G$. Observe that $G$ is a bijective isometry of $S_{c_0}$ onto itself. To prove the plasticity of $S_{c_0}$, it suffices to show $F=G$, or equivalently, to show that $\widetilde{F}:=G^{-1}\circ F$ is the identity map on $S_{c_0}$. We shall establish this fact in what follows. By its definition, $\widetilde{F}$ is a non-expansive bijection from $S_{c_0}$ onto itself. Furthermore, Lemma 2.3 implies that $\widetilde{F}$ acts as the identity on $E$ and satisfies the following properties (also extracted from analogous results for $B_{c_0}$ proved in [6]).

The next lemma is extracted from Lemma 4.7 from [7]. This property was established for $\ell_\infty$ and $c$. However, almost the same argument works for $c_0$.

Main results

Let us consider the set $K$ of points with finite support, such that for any $x \in K$ and any $x^i \in\operatorname{supp}(x)$, $|x^i| = 1$.

Proof. Again, since $\widetilde{F}$ is continuous and $S_{c_0}^{*}$ is dense in $S_{c_0}$, it suffices to consider only elements $x \in S_{c_0}^{*}$, that is, $|\operatorname{supp}(x)| < \infty$. Let us show $\widetilde{F}(x)=x$ gradually increasing the number of nonzero coordinates $x^k$. If $x$ has only one nonzero coordinate, then $x=\pm e_k$ for some $k \in\mathbb{N}$ and there is nothing to prove. If $x$ has two nonzero coordinates, the case with moduli of these coordinates equal to 1 is covered by Lemma 3.1. Now consider $x$ of the following form

with $|x^1|=1$ and $|x^2|<1$. Then $|\widetilde{F}(x)^1|$ is obviously equal to 1. Consider $\widetilde{x}$ with $\widetilde{x}^k=\operatorname{sgn}x^k$ and use Lemma 3.2. We have $\widetilde{F}(x)^2=x^2$.

Analogously, by changing only one coordinate to its sign at each step, we may show $\widetilde{F}(x)=x$ for any $x \in S_{c_0}^{*}$ with arbitrarily many coordinates of modulus 1 and any finite number of coordinates whose moduli are less than 1.

Thus, $\widetilde{F}(x)=x$ holds for all $x \in S_{c_0}^{*}$. Finally, by the density of $S_{c_0}^{*}$ and the continuity of $\widetilde{F}$, we conclude that $\widetilde{F}$ is the identity map on the entire sphere $S_{c_0}$.

Department of Mathematics and Informatics, V. N. Karazin Kharkiv National University, 61022 Kharkiv, Ukraine.

Email address: [email protected]