essental ideal

**Mauritzvdworm** · August 31st 2010, 10:16 AM

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

Show that $I=\{f\in A:f(x)=0 \text{ for all }x\notin O\}$ 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 $g\in A$ such that $gI=\{gf:f\in I\}=\{0\}$ so $g=0$ which implies that I is essential. However, this does not show me anything about O.

**Opalg** · September 1st 2010, 07:55 AM

Originally Posted by Mauritzvdworm

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

Show that $I=\{f\in A:f(x)=0 \text{ for all }x\notin O\}$ 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 $g\in A$ such that $gI=\{gf:f\in I\}=\{0\}$ so $g=0$ which implies that I is essential. However, this does not show me anything about O.

If $O$ is not dense then there is a nonempty open set $P$ disjoint from $O$ . You will need to use something like Urysohn's lemma to get a nonzero function $g\in A$ that vanishes on $O$ . Then $fg = 0$ for all $f \in I$ .

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

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