How to concretely find the characters and pure states of $L^\infty$?

Question

Let $(X, \mathcal X, \mu)$ be a space with measure. Out of curiosity I was trying to understand what the Gelfand-Naimark theorem and Glimm's abstract Stone-Weierstrass theorem give when applied to the $\Bbb C$ $*$-algebra $L^\infty(X,\mu)$. Sadly, I had to give up: given that its elements are not functions with pointwise values, how can I find the pure states and characters of $L^\infty$ (i.e see how they look like concretely)?

Answer 1

Expanding barely on what Shirly said, you cannot expect any explicit image of the characters (and even less the pure states, then). As a C$^*$-algebra, one sees $L^\infty(X)$ as $C(\tilde X)$, where $\tilde X$ is nothing but the space of characters. This is never explicit unless $X$ is finite; even in the infinite discrete case, as Shirly mentioned, $\ell^\infty(\mathbb N)$ is $C(\beta\mathbb N)$, and describing the Stone-Cech compactification is equivalent to describing all free ultrafilters on $\mathbb N$ (which is impossible to do explicitly).

Finally, note that using non-separable C$^*$-algebras as a source for intuition is not a very good idea. There is enough variety and crazyness with separable C$^*$-algebras, that it is better to leave the non-separable ones at rest.

Answer 2

The relevant description is given here (Proposition 2.15).

It boils down to the observation that the sought Gelfand space is the Stone space of the Boolean algebra of measurable sets modulo the null sets.