报告题目：Tingley's problem for positive unit spheres of operator algebras and diametral relations

报 告 人：吴志强教授（南开大学）

报告时间：2024年10月29日下午14:30-15:30

报告地点：腾讯会议860300161

报告摘要: The diametral relation on a bounded metric space X with diameter D is the collection of pairs of elements in X whose distances equal D. In this talk, we consider the bounded metric spaces Pus(A) and P(A) of the positive unit sphere and of the set of projections, respectively, of a complex von Neumann algebra A.

We show that one can use the diametral relation on Pus(A) to identify the subset P(A)\{0} of Pus(A), to determine which projections are central, and to recover the ordering as well as the orthogonality relation on P(A)\{0}.

Let B be another complex von Neumann algebra. We show that if there is an order isomorphism T from P(A)\{0} onto P(B)\{0} preserving the diametral relations, then A and B are real-linear *-isomorphic. In addition, if either A has no type I2 summand, or the map T is metric preserving, then T extends to a real-linear *-isomorphism from A onto B.

Using these, we show that any metric preserving bijection from Pus(A) onto Pus(B) extends to a real-linear *-isomorphism from A onto B. In particular, we obtain an affirmative answer to the Tingley's problem for positive unit spheres of von Neumann algebras.

This talk based on a joint work with Chi-Wai Leung and Ngai-Ching Wong.

报告人简介：吴志强，南开大学陈省身数学研究所教授、博士生导师。主要研究方向为算子代数和泛函分析，在Proceedings of the London Mathematical Society, Journal of Functional Analysis, Mathematische Zeitschrift, Journal of Operator Theory等专业期刊上发表论文数十篇，2005年入选教育部新世纪优秀人才支持计划，主持完成多项国家自然科学基金面上项目。