Are maximal ideals in C*-algebras closed?

Von | Oktober 8, 2022

Surprisingly, the following question is open:

Question: Are maximal ideals in C*-algebras closed?

Let A be a C*-algebra. By an ideal in A we mean a two-sided ideal I \subseteq A that is not necessarily closed. We say that I is a maximal ideal if the only ideals J \subseteq A satisfying I \subseteq J are I and A. An ideal I is proper if I \neq A.

If A is unital, then every maximal ideal is closed. Indeed, in this case, using that a proper ideal does not contain any invertible element of A, and using that the set of invertible elements is open, we see that if I \subseteq A is a proper ideal, then so is its closure \overline{I}. Thus, if I is maximal, then I = \overline{I}, and I 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 I is a non-closed, maximal ideal in a C*-algebra A (if it exists). Then I is dense and therefore contains the Pederson ideal of A. Further, the quotient A/I is a simple \mathbb{C}-algebra. It is easy to see that A/I is radical, that is, \mathrm{rad}(A/I)=A/I, because A/I has no nonzero maximal (modular) left ideals. One can show that I is hereditary (if 0 \leq x \leq y in A, and y belongs to I, then so does x), strongly invariant (if xx^* \in I, then x^*x \in I), and invariant under powers: if a \in I_+ and t > 0, then a^t \in I. For the arguments see this post in mathoverflow.

  1. [1]
    M.C. García, H.G. Dales, Á.R. Palacios, Maximal left ideals in Banach algebras, Bull. London Math. Soc. (2019) 1–15.

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