Duality mapping of Banach space of continuous functions

Question

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 $B=L^p (1<p<+\infty)$, 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 $f\in C_0(X)$.

Many thanks for your kind comments or references in advance.

Answer 1

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.