Related papers: Phase-isometries on the unit sphere of $C(K)$

For a locally compact Hausdorff space $L$, we denote by $C_0(L,\mathbb{R})$ the Banach space of all continuous real-valued functions on $L$ vanishing at infinity, endowed with the supremum norm. In this paper, we prove that every surjective…

For a locally compact Hausdorff space $L$, we denote by $C_0(L,\mathbb{R})$ the Banach space of all continuous real-valued functions on $L$ vanishing at infinity equipped with the supremum norm. We prove that every surjective phase-isometry…

We say that a mapping $f: X \rightarrow Y$ between two real normed spaces is a phase-isometry if it satisfies the functional equation \begin{eqnarray*} \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \quad (x,y\in X).\end{eqnarray*} A…

Let $X$ and $Y$ be real normed spaces and $f \colon X\to Y$ a surjective mapping. Then $f$ satisfies $\{\|f(x)+f(y)\|, \|f(x)-f(y)\|\} = \{\|x+y\|, \|x-y\|\}$, $x,y\in X$, if and only if $f$ is phase equivalent to a surjective linear…

Let $S(C_0(X))^+$ and $S(C_0(Y))^+$ denote the positive parts of the unit spheres of $C_0(X)$ and $C_0(Y)$, where $X$ and $Y$ are locally compact Hausdorff spaces. We prove that every surjective isometry from $S(C_0(X))^+$ onto…

We show that if there exists a Lipschitz homeomorphism $T$ between the nets in the Banach spaces $C(X)$ and $C(Y)$ of continuous real valued functions on compact spaces $X$ and $Y$, then the spaces $X$ and $Y$ are homeomorphic provided…

Let $X$ and $Y$ be completely regular spaces and $E$ and $F$ be Hausdorff topological vector spaces. We call a linear map $T$ from a subspace of $C(X,E)$ into $C(Y,F)$ a \emph{Banach-Stone map} if it has the form $Tf(y) = S_{y}(f(h(y))$ for…

We prove that an onto isometry between unit spheres of finite-dimensional polyhedral Banach spaces extends to a linear isometry of the corresponding spaces.

Motivated by a question posed by Freeman, Oikhberg, Pineau and Taylor, we prove that if $K$ is a compact Hausdorff space with $K^{(\alpha)}\neq\varnothing$, where $2<\alpha<\omega$, then $C[1,\omega^\alpha]$ isometrically embeds into $C(K)$…

In this paper, we show that the $C^1$-differentiability of the norm of a two-dimensional normed space depends only on distances between points of the unit sphere in two different ways. As a consequence, we see that any isometry between the…

Suppose $X$ and $Y$ are locally compact Hausdorff spaces, $E$ and $F$ are Banach spaces and $F$ is strictly convex. We show that every linear isometry $T$ from $C_0(X,E)$ {\em into} $C_0(Y,F)$ is essentially a weighted composition operator…

We extend the existing results on surjective isometries of unit spheres in the Tsirelson space $T\left[\frac{1}{2}, S_1\right]$ to the class $T[\theta,S_{\alpha}]$ for any integer $\theta^{-1} \geq 2$ and $1 \leqslant \alpha < \omega_1$,…

We prove a commutative Gelfand--Naimark type theorem, by showing that the set $C_s(X)$ of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space $X$ (provided with the supremum…

Let $X$ and $Y$ be compact Hausdorff spaces, and let $C(X)$ and $C(Y)$ denote the commutative Banach algebras of all continuous complex-valued functions on $X$ and $Y$, respectively. We study bijective maps $T$ from $C(X)$ onto $C(Y)$ which…

In the recent paper \cite{Hos}, surjective isometries, not necessarily linear, $T: {\rm AC}(X,E) \longrightarrow {\rm AC}(Y,F)$ between vector-valued absolutely continuous functions on compact subsets $X$ and $Y$ of the real line, has been…

Let $A$ be a complex Banach space with a norm $\|f\|=\|f\|_X+\|d(f)\|_Y$ for $f\in A$, where $d$ is a complex linear map from $A$ onto a Banach space $B$, and $\|\cdot\|_K$ represents the supremum norm on a compact Hausdorff space $K$. In…

We show that if $T$ is an isometry (as metric spaces) from an open subgroup of the invertible group $A^{-1}$ of a unital Banach algebra $A$ onto an open subgroup of the invertible group $B^{-1}$ of a unital Banach algebra $B$, then $T$ is…

How can one fully harness the power of physics encoded in relativistic $N$-body phase space? Topologically, phase space is isomorphic to the product space of a simplex and a hypersphere and can be equipped with explicit coordinates and a…

The main result says that every surjective isometry between two ideal Banach function spaces satisfying certain conditions can be presented as a composition of a measurable transformation of a variable and multiplication by a function.

Let $C(X,I)$ be the lattice of all continuous functions on a compact Hausdorff space $X$ with values in the unit interval $I=[0,1]$. We show that for compact Hausdorff spaces $X$ and $Y$ and (not necessarily contain constants) sublattices…