Suppose that $(X,\Sigma,\mu)$ is a measured set with respect to $\sigma$algebra $\Sigma$. Suppose that $L^\infty(X,\mu)$ is the set of all $\mu$equal bounded $\Sigma$measurable functions on $X$. Indeed equally, one may say that $L^\infty(X,\mu)$ is the dual of $L^1(X,\mu)$. What is the spectrum of $(L^\infty(X,\mu),\\cdot\_\infty)$ as a Banach (C^*) algebra? Thank you very much.

Here is a description of the spectrum of $L^\infty([0,1];\mu)$ for an arbitrary Borel measure $\mu$ on $[0,1]$. Consider the following poset, which I call $P$ :
Note that the poset $P$ is filtered: given a finite set $\mathbf X_1, \mathbf X_2, \ldots, \mathbf X_n$ of elements of $P$, there is always a common refinement, i.e., an element $\mathbf X\in P$ such that $\mathbf X\prec \mathbf X_i\\,\forall i$ Given a $\mu$measurable subset $X\subset [0,1]$, let me denote by $X_ \mu \subset [0,1]$ the $\mu$adherence of $X$: $$ X_ \mu:=\{x\in [0,1]: \forall \varepsilon>0\quad \mu(X\cap B_{x,\varepsilon})>0\}, $$ where $B_{x,\varepsilon}$ denotes the ball of radius $\varepsilon$ around the point $x$. Given $\mathbf X=\{X_1,\ldots,X_n\} \in P$, we also write $\mathbf X_ \mu$ for the disjoint union $$\mathbf X_ \mu:=X_1_ \mu\sqcup\ldots\sqcupX_n_ \mu.$$ Note that if $\mathbf X \prec \mathbf Y$, then there is a natural projection map $\mathbf X_ \mu \twoheadrightarrow \mathbf Y_ \mu$. Given the above preliminaries, the spectrum of $L^\infty([0,1];\mu)$ is given by the inverse limit of the functor $P\to Top, \mathbf X\mapsto \mathbf X_ \mu$:


