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 [1]. (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.
- [1]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.