TITLE: NonCommutative Baire theorem
QUESTION [5 upvotes]: 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?

REPLY [3 votes]: This answer (should have been a comment) refers to the second question: 
The kinds of ideal lattices that one can get in a C*-algebra prevent this from happening in general. Take $\mathcal K^\sim$, the unitization of the compact operators (over a separable Hilbert space). The lattice of ideals of $\mathcal K^\sim$ isomorphic to {0,1,2}. The ideal corresponding to the compacts is essential and minimal among the non-zero ideals. So the property of the question cannot hold here. (One can even have the lattice of closed two-sided ideals of a C*-algebra isomorphic to the ordered set [0,1]. See  "A purely infinite AH-algebra and an application to AF-embeddability", by Mikael Rordam.)