Spectra of $C^*$ algebras - MathOverflow

Spectra of $C^*$ algebras

2009-11-11T12:28:07Z

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)?

Answer by Harald Hanche-Olsen for Spectra of $C^*$ algebras

2009-11-11T13:12:43Z

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.

Answer by Matthew Daws for Spectra of $C^*$ algebras

2009-11-11T13:13:18Z

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 . This is the vector space with the unique C*-norm. (Just embed it into C_b(X) for example).

The spectrum of will in general be very large: I don't know any "nice" way of describing it.

Answer by Michael for Spectra of $C^*$ algebras

2009-11-11T14:12:29Z

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.

Answer by Dmitri Pavlov for Spectra of $C^*$ algebras

2009-11-11T15:23:59Z

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, available here: http://gen.lib.rus.ec/get?md5=7F0A9F06741272684D62426E348670B1

Answer by Eric Wofsey for Spectra of $C^*$ algebras

2009-11-11T15:40:25Z

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.