{"ID":22973660,"CreatedAt":"2026-09-17T02:46:45.862280177Z","UpdatedAt":"2026-09-17T02:46:45.862280177Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.18840","arxiv_id":"2609.18840","title":"Maximal Algebraic Ideals in Nonunital $C^*$-Algebras","abstract":"Motivated by Ozawa's question of whether every maximal algebraic two-sided ideal in a $C^*$-algebra must be closed, we study the existence of maximal algebraic two-sided ideals in nonunital $C^*$-algebras. We formulate singular-distribution estimates intrinsically through lower semicontinuous 2-quasitraces and apply them to control algebraic ideal membership. This allows us to develop a novel criterionthe, the admissible quasitracial projection scale, for establishing the nonexistence of maximal ideals in nonunital $C^*$-algebras. This criterion applies to a wide class of simple $C^*$-algebras, including: (i) all $A\\otimes K$ where $A$ is unital, simple, stably finite, $QT_2^1(A)$ nonempty, and the radius of comparison $rc(A)$ is finite, as well as all their hereditary $C^*$-subalgebras whenever $A$ satisfies further assumptions that A is of real rank zero and A has finitely many extreme quasitraces; (ii) all nonunital, simple, separable, stably finite, $Z$-stable $C^*$-algebras A that have an approximate identity consisting of increasing projections $(p_n)$ and for which the simplex $QT_2^1(A,p_1)$ of normalized traces at $p_1$ has finitely many extreme points. We also show that a large class of $C^*$-algebras E constructed from extensions of $C^*$-algebras above such that $E$ still has no maximal ideals. In particular, for these classes, Ozawa's question could be settled in an unexpected manner.","short_abstract":"Motivated by Ozawa's question of whether every maximal algebraic two-sided ideal in a $C^*$-algebra must be closed, we study the existence of maximal algebraic two-sided ideals in nonunital $C^*$-algebras. We formulate singular-distribution estimates intrinsically through lower semicontinuous 2-quasitraces and apply th...","url_abs":"https://arxiv.org/abs/2609.18840","url_pdf":"https://arxiv.org/pdf/2609.18840v1","authors":"[\"Zhichao Liu\",\"Xin Ma\"]","published":"2026-09-16T15:46:23Z","proceeding":"math.OA","tasks":"[\"math.OA\"]","methods":"[]","has_code":false}