# Is every maximal ideal in a C*-algebra always closed?

I wonder if every maximal two-sided (self-adjoint) ideal in a C*-algebra is automatically closed. It is a very basic fact of C*-algebra theory that it holds true for the unital case. In the non-unital case, there certainly exists a non-closed dense ideal but the point is that such an ideal may never be maximal. It is a non-trivial fact that the answer is affirmative for the commutative case [D. Rudd, On isomorphisms between ideals in rings of continuous functions. Trans. Amer. Math. Soc. 159 (1971) 335--353].

One can ask a similar question for maximal *-subalgebras.

• Is the answer known for separable C*-algebras? – Nik Weaver May 25 '17 at 13:59
• @Nik Weaver: Not to my knowledge. I don't know about Banach algebras either. This problem occurred to me sometime ago when I was giving a lecture on functional analysis and C*-algebras. – Narutaka OZAWA May 25 '17 at 23:11
• In the commutative case it is not hard: note first that maximal modular ideals in commutative Banach algebras are maximal and closed. So we need to deal with the non-modular case. However, a maximal ideal $J$ in a commutative Banach algebra $A$ is not modular if and only if $A\cdot A\subseteq J$ and $J$ has co-dimension 1 in $A$. As C*-algebras factor, every maximal ideal in a commutative C*-algebra is modular. – Tomasz Kania May 27 '17 at 16:11
• Hi Tomek. You seem to be using the fact maximal ideals in a CBA always have codimension 1? I was unware of this fact until reading a note of HGD from around 2014, and he seemed to indicate that he did not find this result in the standard sources – Yemon Choi May 29 '17 at 22:11
• For those following along, the HGD note is arxiv.org/abs/1408.3815 – Matthew Daws May 30 '17 at 8:25