Maximal Algebraic Ideals in Nonunital C^*-Algebras

@misc{liu2026maximal,
  title = {{Maximal Algebraic Ideals in Nonunital $C^*$-Algebras}},
  author = {Zhichao Liu and Xin Ma},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.18840},
  eprint = {2609.18840},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18840},
}

TY  - UNPB
TI  - Maximal Algebraic Ideals in Nonunital C^*-Algebras
AU  - Zhichao Liu
AU  - Xin Ma
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.18840
UR  - https://arxiv.org/abs/2609.18840
AB  - In this paper, we investigate the existence of maximal algebraic two-sided ideals in nonunital C^*-algebras. We formulate singular-distribution estimates in terms of lower semicontinuous $2$-quasitraces and introduce a novel criterion, which we call an admissible compact quasitracial triple set. These tools allow us to control membership in algebraic ideals and to prove the nonexistence of maximal ideals in nonunital C^*-algebras under a closed finite-scale face condition. This nonexistence result applies to a broad class of simple C^*-algebras including simple separable stable AF algebras, and nonunital, simple, separable, stably finite, 𝒵-stable C^*-algebras A that admit 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. Moreover, we construct a simple AF algebra whose finite-scale face is dense but not closed and which nevertheless has a maximal dense algebraic ideal. This example yields a negative resolution to a question raised by Ozawa, asking whether every maximal algebraic two-sided ideal in a C^*-algebra must be closed.
ER  -