Spectra of $C^*$ algebras 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)?
 A: 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. 
A: For $L^\infty(X)$, the spectrum is the Stone space of the algebra of measurable sets mod null sets.  This is because a character is determined by what it does on characteristic functions because their span is dense.
A: Your question (especially the first part) is a bit vague, but I'll shoot: A very nice example is provided by Carleson's corona theorem, stating that the unit disk is dense in the spectrum of the Hardy space $H^\infty$ (the bounded holomorphic functions on the unit disk).
As for the spectra of $L^\infty$, I don't think you can ever come up with a concrete example of a character on this space. You actually need the axiom of choice to prove that the spectrum is nonempty. Likewise with the points of the spectrum of $H^\infty$ outside the unit disk.
A: I think, if $X$ is locally compact and Hausdorff, then the spectrum of $C_0(X)$ is just $X$.
You can get the one point compactification of $X$ by looking at the spectrum of the unitisation of $C_0(X)$.  This is the vector space $C_0(X) \oplus \mathbb C$ with the unique $C^*$-norm.  (Just embed it into $C_b(X)$ for example).
The spectrum of $L^\infty(G)$ will in general be very large: I don't know any "nice" way of describing it.
A: The Gelfand representation also works for non-unital commutative C^*-algebras. In this case, it establishes a category equivalence to the category of locally compact Hausdorff spaces with proper maps (implemented by C_0(.) and the spectrum). Hence Matthew's comment, the spectrum of C_0(X) is just X.
A: For another equivalent construction, you can take the set of primitive ideals and endow it with jacobson (or hull-kernel) topology and this space is homeomorphic to the spectrum for commutative things.
This is also the usual definition for the spectrum of a non-commutative $C^*$-algebras. It is analogous to the construction of spectra in algebraic geometry.
