2
$\begingroup$

Let $X = \mathbb{R}^n$ and let $Z$ be a closed subspace of $X$. The inclusion map $i: Z \hookrightarrow X$ induces a map $C_0(X) \to C_0(Z)$ (via pullback). My question is simple: what is the kernel? The answer should either be $C_0(X - Z)$ or $C(X - Z)$ and I think I know which one is correct. But I need a sanity check, so I thought I would post the question here.

  • 5
    Can you tell us what you mean by $C_0$?2010-08-26
  • 1
    Paul, please edit the question making explicit what $C_0$ is.2010-11-13
  • 0
    Can you tell us what do you mean by subspace? Linear or topological? With finite dimensions linear spaces, every subspace is closed.2010-11-13

2 Answers 2

2

If $C_0$ denotes functions vanishing at infinity, then it is $C_0(X\setminus Z)$. The other candidate is much too big to be an ideal in $C_0(X)$.

If, on the other hand, $C_0$ denotes compactly supported functions, then Robin Chapman's answer applies.

  • 1
    This is correct. $C(X\setminus Z)$ isn't even a Banach algebra, and it will contain many elements that don't come from $C_0(X)$, the most obvious being its unbounded elements.2010-08-27
  • 0
    Your interpretation of $C_0$ is the one I had in mind. I kept freaking myself out about what happens near the boundary for no good reason, but it was all a bunch of pointless fuss. Thanks to both of you for helping me get grounded in reality.2010-08-31
2

I presume $C_0(X)$ means compactly supported continuous functions on $X$. So you are asking which compactly supported continuous functions on $X$ which vanish on $Z$. Now these may not be compactly supported on $X-Z$ for there may be a point $p$ in $Z$ and a sequence of points $(p_n)$ in converging to $X$ with $f(p_n)\ne0$ but $f(p)=0$. On the other hand there will be continuous functions on $X-Z$ which don't converge to zero on the boundary. So neither $C(X-Z)$ nor $C_0(X-Z)$ is correct!

  • 1
    I think $C_0(X)$ means continuous functions vanishing at infinity, that is, which are arbitrarily small outside of sufficiently large compact subsets.2010-08-26