Z-stability and finite dimensional approximation

林华新  (Shanghai Institute for Mathematics and Interdisciplinary Sciences)

10:30-11:30, October 28, 2026   闵行校区数学楼 102

Abstract:

We introduce the notion of essential tracial rank at most one for simple C*-algebras. We show that separable simple C*-algebras with essential tracial rank at most one have several regularity properties, including stable rank one, strict comparison, and tracial approximate divisibility. A finite amenable simple C*-algebra with this property is Z-stable. We also prove the converse. Thus, this property gives a new characterization of separable simple finite amenable Z-stable C*-algebras and shows that these algebras admit a one-dimensional tracial approximation. In the unital real-rank-zero case, this approximation can be strengthened to show that the algebra has tracial rank zero. This completes a long-standing project. As a byproduct, we prove that every tracial state on a unital separable simple finite amenable C*-algebra is quasidiagonal and that the algebra itself is quasidiagonal. Consequently, every separable simple finite amenable C*-algebra of finite nuclear dimension has finite decomposition rank. None of these results requires the UCT.

About the speaker:

林华新，上海数学与交叉科学研究院教授、华东师范大学退休教授、华东师范大学算子代数研究中心创始人，美国数学会首届会士，国际算子代数领域的领军学者。 林教授在C*-代数分类理论方面取得了突破性成果，解决了著名的Halmos问题，创立了具有里程碑意义的迹秩理论，并与合作者完成了“Elliott分类纲领”，推动了算子代数领域的重大进展。其研究成果发表于Annals of Mathematics、Inventiones Mathematicae、PNAS等国际顶级期刊。林教授曾获2005年度上海市科学技术进步一等奖，并于2023年荣获首届国际基础科学大会前沿科学奖，曾主持美国国家科学基金会十余项研究项目，以深厚的理论造诣和卓越的学术领导力持续引领该领域的发展。

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