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

• DOI: 10.1007/s43034-021-00141-w
• Publication Date: 2021-08-12T10:03:01Z
• Authors: Longfa Sun
• Keywords: Surjective function; Homeomorphism (graph theory); Isometry (Riemannian geometry); Infimum and supremum

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