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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 4th 2025
   • (edited Apr 4th 2025)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIt is being pointed out to me by email that this entry says about the algebra $C_0(X)$ of functions vanishing at infinity that: > $C_0(X)$ is no longer a Banach space (due to [revision 1](https://ncatlab.org/nlab/revision/Stone-Weierstrass+theorem/1) by Todd Trimble, way back in October 2009) This seems odd, as $C_0(X)$ is a standard example of a Banach space, unless something else is meant here. It seems nothing in the entry depends on this side-remark, so that it may be worth deleting. <a href="https://ncatlab.org/nlab/revision/diff/Stone-Weierstrass+theorem/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/Stone-Weierstrass+theorem/11">v11</a>, <a href="https://ncatlab.org/nlab/show/Stone-Weierstrass+theorem">current</a>

   It is being pointed out to me by email that this entry says about the algebra $C_0(X)$ of functions vanishing at infinity that:

   $C_0(X)$ is no longer a Banach space

   (due to revision 1 by Todd Trimble, way back in October 2009)

   This seems odd, as $C_0(X)$ is a standard example of a Banach space, unless something else is meant here.

   It seems nothing in the entry depends on this side-remark, so that it may be worth deleting.

   diff, v11, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 8th 2025
   • PermaLink
   Author: Urs
   Format: MarkdownItexHave deleted the parenthesis starting by saying that $C_0(X)$ is not a Banach space. <a href="https://ncatlab.org/nlab/revision/diff/Stone-Weierstrass+theorem/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/Stone-Weierstrass+theorem/12">v12</a>, <a href="https://ncatlab.org/nlab/show/Stone-Weierstrass+theorem">current</a>

   Have deleted the parenthesis starting by saying that $C_0(X)$ is not a Banach space.

   diff, v12, current

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 26th 2025
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexNo idea what I had in mind!

   No idea what I had in mind!

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 26th 2025
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for getting back on that point. In any case it would be nice if we had a page or paragraph on the $C^\ast$ algebras $C_0(-)$, to link to for more information. Maybe something could be added to the page *[[vanishing at infinity]]*.

   Thanks for getting back on that point.

   In any case it would be nice if we had a page or paragraph on the $C^\ast$ algebras $C_0(-)$, to link to for more information. Maybe something could be added to the page vanishing at infinity.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 26th 2025
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexRight, that would be appropriate. As penance for my 2009 sin, I'd be happy to write something up a little later today. The immediate thought is to identify $C_0(X)$ with the kernel $K$ of the map $ev_\infty: C(\bar{X}) \to \mathbb{C}$ where $\bar{X}$ is the one-point compactification and $\infty$ is the adjoined point. This kernel is closed in the Banach space $C(\bar{X})$, hence is itself a Banach space, and the $C^\ast$-algebra structure is not far behind.

   Right, that would be appropriate. As penance for my 2009 sin, I’d be happy to write something up a little later today. The immediate thought is to identify $C_0(X)$ with the kernel $K$ of the map $ev_\infty: C(\bar{X}) \to \mathbb{C}$ where $\bar{X}$ is the one-point compactification and $\infty$ is the adjoined point. This kernel is closed in the Banach space $C(\bar{X})$, hence is itself a Banach space, and the $C^\ast$-algebra structure is not far behind.

6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 26th 2025
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexRe 4: I wrote a paragraph over at [[vanishing at infinity]], and linked to it from the current page here.

   Re 4: I wrote a paragraph over at vanishing at infinity, and linked to it from the current page here.

7. • CommentRowNumber7.
   • CommentAuthorDean
   • CommentTimeJun 30th 2025
   • (edited Jun 30th 2025)
   • PermaLink
   Author: Dean
   Format: MarkdownItexMaybe the mistake could be because of the boundedness in GNS-construction. I thought I read the same thing elsewhere.

   Maybe the mistake could be because of the boundedness in GNS-construction. I thought I read the same thing elsewhere.