What is the commutative analogue of a $C^*$ - subalgebra?

Making use of the duality in between in your area portable Hausdorff rooms and also commutative $C^*$ - algebras one can list a vocabulary checklist converting topological ideas pertaining to an in your area portable Hausdorff room $X$ right into algebraic ideas ragarding its ring of features $C_0(X)$ (see Wegge - Olsen's publication, as an example). As an example, we have the adhering to documents : \begin{align*} \text{open subset of $X$}\quad &\longleftrightarrow\quad\text{ideal in $C_0(X)$}\newline \text{dense open subset of $X$}\quad &\longleftrightarrow\quad\text{essential ideal in $C_0(X)$}\newline \text{closed subset of $X$}\quad &\longleftrightarrow\quad\text{quotient of $C_0(X)$}\newline \text{locally closed subset of $X$}\quad &\longleftrightarrow\quad\text{subquotient of $C_0(X)$}\newline \text{???}\quad &\longleftrightarrow\quad\text{$C^*$-subalgebra in $C_0(X)$} \end{align*} By excellent I constantly suggest a 2 - sided shut (and also therefore self - adjoint) perfect.

Well, I can not fairly see just how to reconvert a $C^*$ - subalgebra in $C_0(X)$ right into something topological entailing just the room $X$. Can you think of something convenient?

Instance : A straightforward instance of a subalgebra of a commutative $C^*$ - algebra not being a perfect is $$ \mathbb C\cdot(1,1)\subset \mathbb C\oplus\mathbb C. $$

(Alternatively, we can think of this inquiry within the duality of affine algebraic selections and also finitely created commutative lowered algebras or perhaps within the duality in between affine systems and also commutative rings.)

Edit : Since I was not entirely pleased by the feedback I obtained below, I reposted this inquiry on MO.

2019-05-09 11:43:14
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Answers: 1

Roughly, the solution will certainly be that shut C * - subalgebras will certainly represent quotient rooms (using pull - rear of features). In your instance, the quotient map is one which recognizes both factors right into a solitary factor. I have not analyzed, though, whether this is an entirely proper declaration as it stands, or whether one needs to add added cautions.

[Included in address an inquiry in remarks : ] The suggestion is that if $X$ surjects onto $Y$ after that we get a shot $C_0(Y) \to C_0(X)$, and also alternatively.

[Added conversation included after extra assumed : ] Let me claim something concerning the similar scenario in algebraic geometry, where I am extra comfy with the technological concerns :

Affine algebraic collections over $\mathbb C$ represent limited type lowered $\mathbb C$ - algebras Offering an incorporation $A \hookrightarrow B$ of limited type lowered $\mathbb C$ - algebras. represents offering a map $X \to Y$ of algebraic collections which is leading, i.e. the photo is thick.

Currently in your set - up : if $X \to Y$ is a map of in your area portable Hausdorff rooms with thick photo, however the map $C_0(X) \to C_0(Y)$ will certainly be injective ; so I could have been also rash when I insisted that we get a surjective map. On the various other hand, probably the photo of $C_0(X) \to C_0(Y)$ will certainly not be enclosed this degree of generalization ; it's a while given that I've assumed meticulously concerning these type of points, so I do not assume I can claim extra now with any kind of assurance.

Specifically, I'm not so made use of to operating in the instance of rings without device, so my pointer has even more opportunity to be proper in case when the rooms are portable. So probably it would certainly be most convenient to think of the instance when $X$ and also $Y$ are portable first ; note after that a map with thick photo will instantly be surjective, therefore this instance could be less complex to recognize consequently also. (In reality, thinking of your instance of a perfect that you state in remarks, it could be less complicated to pass to one - factor compactifications - - - and also hence add a device - - - prior to continuing. Due to the fact that without a doubt I assume that in the excellent instance, what will certainly take place is that we will certainly get a map from the 1 factor compactification of the room to the one factor compactification of the open set which squashes the enhance of the open put down to the factor at infinity, specifically as you recommend in your comment.)

2019-05-10 08:21:56