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Using the duality between locally compact Hausdorff spaces and commutative $C^*$-algebras one can write down a vocabulary list translating topological notions regarding a locally compact Hausdorff space $X$ into algebraic notions regarding its ring of functions $C_0(X)$ (see Wegge-Olsen's book, for instance). For example, we have the following correspondences: $$ \;\;\;\text{open subset of $X$}\quad \longleftrightarrow\quad\text{ideal in $C_0(X)$} $$ $$ \;\;\;\;\;\quad\text{dense open subset of $X$}\quad \longleftrightarrow\quad\text{essential ideal in $C_0(X)$} $$ $$ \;\;\;\quad\text{closed subset of $X$}\quad \longleftrightarrow\quad\text{quotient of $C_0(X)$} $$ $$ \text{locally closed subset of $X$}\quad \longleftrightarrow\quad\text{subquotient of $C_0(X)$} $$ $$ \;\;\;\quad\qquad\qquad\qquad\qquad\text{???}\qquad\qquad \longleftrightarrow\quad\text{$C^*$-subalgebra in $C_0(X)$} $$ By ideal I always mean a two-sided closed (and hence self-adjoint) ideal.

Well, I can't quite see how to reconvert a $C^*$-subalgebra in $C_0(X)$ into something topological involving only the space $X$ (and some data describing the subalgebra in topological terms). Can you come up with something handy?

Example: A simple example of a subalgebra of a commutative $C^*$-algebra not being an ideal is $$ \mathbb C\cdot(1,1)\subset \mathbb C\oplus\mathbb C. $$

First attempts: Instead of talking about a subalgebra, we should probably talk about the injective $^* $-homomorphism given by the inclusion of this subalgebra. But is this inclusion proper (i.e., does it preserve approximate units) in general? Well, at least when we restrict to compact spaces. Then an injective $^* $-homomorphism $C(Y)\to C(X)$ will induce a surjective continuous map $X\to Y$. How to proceed?

Remark: Alternatively, we could think about this question within the duality of affine algebraic varieties and finitely generated commutative reduced algebras or even within the duality between affine schemes and commutative rings.

Disclaimer: I posted this question yesterday on MSE. I also got an interesting answer. However, I'm not yet fully satisfied. If I violate any policy by reposting the question here, please tell me about it.

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I think your MSE answer looks pretty good. –  Steve Huntsman Aug 13 '10 at 16:57
My crunchpoint is the step from the surjection $X\to Y$ to some quotient. I don't see how this should work without losing something. –  Rasmus Bentmann Aug 13 '10 at 17:02
You probably meant "violate" in your last sentence, but I like "validate" better. In any case, no, I think reposting a research-level question from MSE to MO is appropriate, and I think this question is "research-level", whatever that means. –  Theo Johnson-Freyd Aug 16 '10 at 5:28
@Theo Johnson-Freyd: Yes, thank you -- I meant violate. –  Rasmus Bentmann Aug 16 '10 at 9:17
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1 Answer

up vote 12 down vote accepted

Let $A$ be a commutative $C^\star$-algebra and $B$ a $C^\star$-subalgebra (in general, this will not preserve approximate units as the example ${\mathbb C} \oplus 0 \subset {\mathbb C} \oplus{\mathbb C}$ shows). This gives also rise to an inclusion of the unitalization $B^+$ into $A^+$. Now you are in the realm of unital $C^\star$-algebras and your own remark applies. If $A$ was isomorphic to the algebra of continuous functions on the locally compact space $X$, then $A^+$ is the algebra of continuous functions on the one-point compactification $X^+$ of $X$. Now, $B^+ \cong C(Y)$, where $Y$ is some quotient of the one-point compactification of $X$. And again, $Y$ is the one-point compactification of some space locally compact space $Z$ such that $B = C_0(Z)$. What is the relation between $Z$ and $X$? Well, the continuous quotient map $f \colon X^+ \to Z^+=Y$ gives rise to a canonical open subset $U \subset X$, which arises as the pre-image of $Z$. Moreover, one gets a quotient map $f|_U \colon U \to Z$ and $f|_U$ is easily seen to be proper. This is everything that can be said.

(Conversely, let $U \subset X$ be open and $g \colon U \to Z$ be a proper surjection. Then $C_0(Z) \to C_0(U) \subset C_0(X)$ gives rise to a sub-algebra.)

This is explained in more detail in the book by Higson and Roe (see here).

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@Andreas: Thank you for your answer. So, if I understand correctly, the simplest replacement for my ??? above is "an open subset of $X$ together with a proper surjection from this open subset to some locally compact space". –  Rasmus Bentmann Aug 13 '10 at 17:21
Right, up to some obvious notion of equivalence. –  Andreas Thom Aug 13 '10 at 17:58
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