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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