The classical Baire theorem says that the intersection of a sequence of open dense subsets of $X$, is dense, if the space is compact Hausdorff. In the language of $C^{*}$ algebras this is equivalents to the following:

Assume that $I_{n},s$ are a sequence of essential ideals in a commutative unital $C^{*}$ algebra $A$. Assume that $J$ is an arbitrary nontrivial ideal.Then there is a maximal ideal $L$ which neither contain $J$, nor contains $I_{n}$. (no $I_{n}$ is contained in $L$).

This is a motivation to ask:

Is the above statement true for a commutative unital semisimple Banach algebra?

Is the above statement true for a unital non commutative $C^{*}$ algebra, provided we replace the "maximal ideals" by primitive Ideals?

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