dual space of space of space of bounded continuous maps

Question

$T: C_b(X, \mathbb R)\to C_b(X,\mathbb R)$ a linear map where $X$ is complete, separable, locally compact metric space. $C_b(X,\mathbb R)$ is vector space of all bounded continuous functions. Can I say that $T^*: M_1(X)\to M_1(X)$ where $M(X)$ is space of all probability measures on $X$? Thanks!

Answer 1

No. Even for locally compact $X$, the dual of $C_b(X)$ is larger than the space of (regular Borel) measures.

Say $X=\Bbb N$, so $C_b(X)=\ell_\infty$. Let $\Lambda\in\ell_\infty^*$ be a Banach limit, which is to say that $\Lambda x=\lim_{j\to\infty}x_j$ for all $x\in\ell_\infty$ such that the limit exists. Then $\Lambda$ is not given by a measure on $X$.

Note $\Bbb N$ is certainly locally compact and separable; it's also complete in the standard metric $|n-m|$.

In fact one can give an analogous example if $X$ is any non-compact locally compact Hausdorff space. A bit of Banach algebra stuff shows that $C_b(X)$ is isometrically isomorphic to $C(K)$, where $K$ is a certain compact Hausdorff space containing $X$ as a dense subset. Any measure on $K$ which is not supported on $X$ gives an example of a bounded linear functional on $C_b(X)$ which does not arise from a measure on $X$.