On non-linear $\varepsilon$-isometries between the positive cones of certain continuous function spaces

Abstract

Let X, Y be two $w^*$-almost smooth Banach spaces, $C(B(X^*),w^*)$ be the Banach space of all continuous real-valued functions on $B(X^*)$ endowed with the supremum norm and $C_+(B(X^*),w^*)$ be the positive cone of $C(B(X^*),w^*)$. In this paper, we show that if $F: C_+(B(X^*),w^*)\rightarrow C_+(B(Y^*),w^*)$ is a standard almost surjective $\varepsilon$-isometry, then there exists a homeomorphism $\tau : B(X^*)\rightarrow B(Y^*)$ in the $w^*$-topology such that for any $x^*\in B(X^*)$, we have

$$\begin{aligned} |\langle \delta _{x^*}, f\rangle -\langle \delta _{\tau (x^*)}, F(f)\rangle |\le 2\varepsilon ,\quad \text{for all } f\in C_+(B(X^*),w^*). \end{aligned}$$

As its application, we show that if $U:C(B(X^*),w^*)\rightarrow C(B(Y^*),w^*)$ is the canonical linear surjective isometry induced by the homeomorphism $\gamma =\tau ^{-1}:B(Y^*)\rightarrow B(X^*)$ in the $w^*$-topology, then

$$\begin{aligned} \Vert F(f)-U(f)\Vert \le 2\varepsilon , \quad \text{for all }f\in C_+(B(X^*),w^*). \end{aligned}$$