Gelfand-Naimark structure theorem for $C^* $ algebras gives a canonical isometric * isomorphism between any commutative unital $C^* $ algebra $A$ and the algebra of continuous complex-valued functions on $A$^. This is the spectrum (or structure space) of $A$, i.e. the non-zero multiplicative linear continuous functionals with the topology of pointwise convergence (alias weak*), which is compact and hausdorff. Apart from the easy case $A = C(X)$, with $X$ compact hausdorff, for which $A$^ is $X$ itself, there are a lot of non trivial and not immediately visible examples of spectra, for example:
If $X$ loc. compact hausdorff $A = C_b(X)$ (continuous and bounded functions with uniform topology) is a $C^*$ algebra. If X is non compact then A^ cannot be $X$ and is in fact $\beta X$, the Stone-Cech compactification of $X$.
If $X$ is loc. compact hausdorff and you take $C_0(X)$, then you get another compactification of $X$.
If instead you simply take $C(X)$ for $X$ compact non-hausdorff you get a natural "hausdorfization" of $X$.
I'm particularly interested in the existence of other constructions which can be described by gelfand theory as above. I mean to associate functorially to each space (in an appropriate subcategory of Top, maybe not full) a $C^*$ algebra and then to look at its spectra.
A related question: what are the spectra of $L^\infty(R)$, and similar algebras (maybe $L^\infty(G)$, G loc. compact group with haar measure)?
Best Answer
The spectrum of $L^\infty(R)$ is the hyperstonean space associated with the measurable space R. More information can be found in Takesaki's Theory of Operator Algebras I, Chapter III, Section 1.