Surprisingly, the following question is open:
Question: Are maximal ideals in C*-algebras closed?
Let be a C*-algebra. By an ideal in we mean a two-sided ideal that is not necessarily closed. We say that is a maximal ideal if the only ideals satisfying are and . An ideal is proper if .
If is unital, then every maximal ideal is closed. Indeed, in this case, using that a proper ideal does not contain any invertible element of , and using that the set of invertible elements is open, we see that if is a proper ideal, then so is its closure . Thus, if is maximal, then , and is closed.
Further, it is known that every maximal left ideal in a C*-algebra is closed; see for example Proposition 3.5 in . (This even holds in Banach algebras admitting a bounded approximate unit.) In commutative C*-algebras, every maximal ideal is also a maximal left ideal and therefore closed.
Assume is a non-closed, maximal ideal in a C*-algebra (if it exists). Then is dense and therefore contains the Pederson ideal of . Further, the quotient is a simple -algebra. It is easy to see that is radical, that is, , because has no nonzero maximal (modular) left ideals. One can show that is hereditary (if in , and belongs to , then so does ), strongly invariant (if , then ), and invariant under powers: if and , then . For the arguments see this post in mathoverflow.
- M.C. García, H.G. Dales, Á.R. Palacios, Maximal left ideals in Banach algebras, Bull. London Math. Soc. (2019) 1–15. https://doi.org/10.1112/blms.12290.