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Is it true that in a unital C*-algebra $A$ every closed left ideal $L\subset A$ is an intersection of all maximal left ideals which contain $L$? I know that $L$ is the left-kernel of some state but I am not sure whether this helps.

Edit: This is true. See Banach Algebras and the General Theory of *-Algebras: Volume 2, *-Algebras; Theorem 10.5.2(b).

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Rather than delete the question you should add the answer as an answer! (And, after a couple of days, accept it) –  Mariano Suárez-Alvarez Sep 4 '12 at 18:19

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This is true. See Banach Algebras and the General Theory of *-Algebras: Volume 2, *-Algebras; Theorem 10.5.2(b).

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