dual of $L^\infty$ or $l^{\infty}$ spaces and characters of these spaces

Question

Let $\nu$ be a $\sigma$-additive probability measure on some standard Borel space $(X,\Sigma)$. By Gelfand's transform or by Stone-Cech compactification $L^{\infty}(X,\nu)$ is isomorphic to $C(\beta(X))$ where $\beta(X)$ is the compact space of ultrafilters on the complete boolean algebra $\textsf{B}_\nu=\Sigma/\mathcal{Null}_\nu$ and $\mathcal{Null}_\nu$ denotes the $\sigma$-ideal of null $\nu$-measure zero sets. In particular the characters (i.e. the homomorphisms of $L^{\infty}(X,\nu)$ with $\mathbb{C}$) can be identified with the ultra filters $G$ on $\mathsf{B}_\nu$. On the other hand $(L^{\infty}(X,\nu))^*$ can be represented as the space of finitely additive signed measure $\lambda$ which are absolutely continuous with respect to $\nu$ (see answer to mathstackexchange question Dual space of L infty space). What is the natural map which identifies the characters as a compact subset in the $*$-weak topology on $(L^{\infty}(X,\nu))^*$ (i.e. the topology generated by the linear functionals on $(L^{\infty}(X,\nu))^*$ given by $L^\infty$-functions)? What kind of signed measures are these characters?

Answer 1

The multiplicative ones - those for which $\intop fg d \mu = \intop f d \mu \cdot \intop g d \mu$.

And from this all sorts of things follow that make them act like $\delta$-measures: e.g. they are positive and they respect the continuous functional calculus - $\intop F(f) d \mu = F\left( \intop f d \mu \right)$.