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Thread: essental ideal

  1. #1
    Member Mauritzvdworm's Avatar
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    essental ideal

    Let $\displaystyle A=C_0(X)$ where X is some locally compact Hausdorff space and let $\displaystyle O\subset X$ be open.

    Show that $\displaystyle I=\{f\in A:f(x)=0 \text{ for all }x\notin O\}$ is an essential ideal in A if and only if O is dense in X

    It is easy to see that I is an ideal in A

    My plan involved choosing a specific $\displaystyle g\in A$ such that $\displaystyle gI=\{gf:f\in I\}=\{0\}$ so $\displaystyle g=0$ which implies that I is essential. However, this does not show me anything about O.
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  2. #2
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    Quote Originally Posted by Mauritzvdworm View Post
    Let $\displaystyle A=C_0(X)$ where X is some locally compact Hausdorff space and let $\displaystyle O\subset X$ be open.

    Show that $\displaystyle I=\{f\in A:f(x)=0 \text{ for all }x\notin O\}$ is an essential ideal in A if and only if O is dense in X

    It is easy to see that I is an ideal in A

    My plan involved choosing a specific $\displaystyle g\in A$ such that $\displaystyle gI=\{gf:f\in I\}=\{0\}$ so $\displaystyle g=0$ which implies that I is essential. However, this does not show me anything about O.
    If $\displaystyle O$ is not dense then there is a nonempty open set $\displaystyle P$ disjoint from $\displaystyle O$. You will need to use something like Urysohn's lemma to get a nonzero function $\displaystyle g\in A$ that vanishes on $\displaystyle O$. Then $\displaystyle fg = 0$ for all $\displaystyle f \in I$.

    For the converse, if $\displaystyle O$ is dense in $\displaystyle X$ then any nonzero function $\displaystyle g\in A$ must be nonzero at some point $\displaystyle x_0\in O$. Use Urysohn's lemma again, to show that there is a function $\displaystyle f\in I$ with $\displaystyle f(x_0)\ne0$. That will show that $\displaystyle I$ is essential.
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