非单位 $C^*$-代数中的极大代数理想
Maximal Algebraic Ideals in Nonunital $C^*$-Algebras
- Dalian University of Technology（大连理工大学）
- Harbin Institute of Technology（哈尔滨工业大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究非单位$C^*$-代数中极大代数双边理想的存在性，提出容许拟迹投影尺度准则，证明多类单$C^*$-代数及其扩张无极大理想，回应Ozawa问题。
AI中文摘要：
受Ozawa关于$C^*$-代数中每个极大代数双边理想是否必须闭的问题的启发，我们研究了非单位$C^*$-代数中极大代数双边理想的存在性。我们通过下半连续2-拟迹内在地表述奇异分布估计，并将其应用于控制代数理想的隶属关系。这使我们能够发展一个新的准则——容许拟迹投影尺度，用于建立非单位$C^*$-代数中极大理想的不存在性。该准则适用于一大类单$C^*$-代数，包括：(i) 所有$A\otimes K$，其中$A$是单位的、单的、稳定有限的，$QT_2^1(A)$非空，且比较半径$rc(A)$有限，以及当$A$满足进一步假设（$A$具有实秩零且$A$具有有限多个极端拟迹）时，它们的所有遗传$C^*$-子代数；(ii) 所有非单位的、单的、可分的、稳定有限的、$Z$-稳定的$C^*$-代数$A$，这些代数具有由递增投影$(p_n)$组成的近似单位，并且使得在$p_1$处的归一化迹的单纯形$QT_2^1(A,p_1)$具有有限多个极端点。我们还证明，由上述$C^*$-代数的扩张构造的一大类$C^*$-代数$E$仍然没有极大理想。特别地，对于这些类，Ozawa的问题可以以出人意料的方式得到解决。
英文摘要：
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.
