Ozawa’s closedness question for maximal algebraic ideals

Determine whether every maximal algebraic two-sided ideal in an arbitrary C*-algebra is norm closed.

Background

The paper distinguishes algebraic two-sided ideals, which need not be norm closed, from closed ideals in C*-algebras. It recalls that maximal algebraic ideals are closed in several settings, including unital C*-algebras and algebras of the form C_0(X,A) with A unital.

The authors identify the general case posed by Ozawa as unresolved. Their results provide partial evidence toward an affirmative answer by proving that broad classes of simple nonunital C*-algebras have no maximal algebraic ideals at all, rather than producing nonclosed maximal ideals.