Existence of maximal ideals in nonunital C*-algebras

Determine whether every nonunital C*-algebra admits a maximal algebraic two-sided ideal.

Background

The paper explains that, unlike the unital case, Zorn’s lemma does not directly guarantee a maximal algebraic ideal in a nonunital C*-algebra because an increasing chain of proper ideals may exhaust the algebra. The existence issue is therefore logically prior to asking whether maximal ideals are closed.

The authors prove nonexistence results for substantial classes, including many simple stable and nonstable C*-algebras and certain nonsimple extensions, but do not resolve the question for all nonunital C*-algebras.

References

For non-simple $C*$-algebras, by contrast, proper closed ideals may exist, yet it remains unknown whether a maximal ideal---closed or not---always exists. The existence question is therefore fundamental: if no such ideal exists, then the issue of whether it is closed becomes vacuous, and Ozawa's question has no object to which it can apply. This naturally leads to the following question, which may be regarded as a preliminary counterpart to Ozawa's. Does every nonunital $C*$-algebra admit a maximal ideal?

— Maximal Algebraic Ideals in Nonunital $C^*$-Algebras  (2609.18840 - Liu et al., 16 Sep 2026) in Question 1.1, Section 1, Introduction