# Duality mapping of Banach space of continuous functions

Let B* be the dual space of a Banach space B. Recall the definition of the duality mapping (set-valued mapping) $$J: B\to 2^{B^*}, J(f):=\{L\in B^*: L(f)=||L||||f|, ||L||=||f||\}.$$ For instance, if $$B=L^p (1, then $$J(f)=\frac{\overline{f}|f|^{p-1}}{||f||_{L^p}^{p-2}},\ 0\ne f\in L^p.$$ My question is about the characterization of the duality mapping of Banach space $$C_0(X)$$ of all continuous functions vanishing at infinity on a locally compact Hausdorff space $$X$$. The space $$C_0(X)$$ is endowed with the maximum norm $$||\cdot||_{C_0(X)}$$. The dual space of $$C_0(X)$$ is the Banach space $$M(X)$$ of all the complex-valued regular Borel measures on $$X$$ with a bounded total variation [Rudin, Real and Complex Analysis, 1987, Theorem 6.19]. In other words, every continuous functional $$T$$ on $$C_0(X)$$ is represented by a $$\mu\in M(X)$$ in the sense that $$T(f)=\int_X f(x)d\mu(x),\mbox{ for all }f\in C_0(X).$$

For every $$f\in C_0(X)$$, there exists a maximum. The duality mapping $$J(f)$$ must be closely related the set of maximum points of the continuous function $$f$$, i.e, $$A=\{x\in X: |f(x)|=||f||_{C_0(X)}\}.$$

My question is how to characterize $$J(f)$$ for any $$f\in C_0(X)$$.

Let's suppose that $$\|f\|=1$$. Let $$P$$ denote the set of probability measures supported on the compact set $$A$$. Then $$J(f)$$ consists of all measures of the form $$h\,d\mu$$, $$\mu\in P$$, where, for all $$x$$ in the support of $$\mu$$, $$|h(x)|=1$$ and $$f(x)h(x)\ge 0$$. In the real case that means that $$h(x)$$ and $$f(x)$$ have the same sign, and $$h$$ is the difference of two indicator functions.