报告题目： On a variant of Tingley's problem for some function spaces

报 告 人：吴志强 教授（南开大学）

时    间：2020年12月30号（星期三） 上午10:00-11:00

地    点：腾讯会议 775959124

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，先后在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，先后主持完成国家自然科学基金面上项目4项，目前正主持国家自然科学基金面上项目1项。

报告摘要:

   Let $(\Omega, \mathfrak{A}, \mu)$ and $(\Gamma, \mathfrak{B}, \nu)$ be two arbitrary measure spaces, and $p\in [1,\infty]$. Set $$S(L^p(\mu))_+:= \{f\in L^p(\mu): \|f\|_p =1; f\geq 0\ \mu\text{-a.e.} \},$$ that is, the positive part of the unit sphere of $L^p(\mu)$. We show  that every surjective isometry $\Phi: S(L^p(\mu))_+\to S(L^p(\nu))_+$ can be extended (necessarily uniquely) to an isometric order isomorphism from $L^p(\mu)$ onto $L^p(\nu)$. A Lamperti form, i.e., a weighted composition like form, of $\Phi$ is provided, when $(\Gamma, \mathfrak{B}, \nu)$ is localizable (in particular, when it is $\sigma$-finite).
   On the other hand, we show that for compact Hausdorff spaces $X$ and $Y$, if $\Phi$  is a surjective isometry from the positive part of the unit sphere of $C(X)$ to that of $C(Y)$, then there is a homeomorphism $\tau:Y\to X$ satisfying $\Phi(f)(y) = f(\tau(y))$ for $f\in S(C(X))_+$ and $y\in Y$. [Joint work with Chi-Wai Leung and Ngai-Ching Wong]