Abstract
Let X and Y be locally compact Hausdorff spaces. In this paper we study surjections \(T: A \longrightarrow B\) between certain subsets A and B of \(C_0(X)\) and \(C_0(Y)\), respectively, satisfying the norm condition \(\Vert \varphi (Tf, Tg)\Vert _Y=\Vert \varphi (f,g)\Vert _X\), \(f,g \in A\), for some continuous function \(\varphi : {\mathbb {C}}\times {\mathbb {C}}\longrightarrow {\mathbb {R}}^+\). Here \(\Vert \cdot \Vert _X\) and \(\Vert \cdot \Vert _Y\) denote the supremum norms on \(C_0(X)\) and \(C_0(Y)\), respectively. We show that if A and B are (positive parts of) subspaces or multiplicative subsets, then T is a composition operator (in modulus) inducing a homeomorphism between strong boundary points of A and B. Our results generalize the recent results concerning multiplicatively norm preserving maps, as well as, norm additive in modulus maps between function algebras to more general cases.
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Jafarzadeh, B., Sady, F. Generalized norm preserving maps between subsets of continuous functions. Positivity 23, 111–123 (2019). https://doi.org/10.1007/s11117-018-0597-y
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DOI: https://doi.org/10.1007/s11117-018-0597-y