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Math Help - essental ideal

  1. #1
    Member Mauritzvdworm's Avatar
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    essental ideal

    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.
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  2. #2
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    Opalg's Avatar
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    Quote Originally Posted by Mauritzvdworm View Post
    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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