报告题目： From linear to nonlinear bounded approximation property for operators and Banach spaces

报 告 人： 刘锐 教授 (南开大学)

报告时间： 2022年5月30日 上午10:00-11:00

报告地点：腾讯会议 会议ID：178-286-240

邀 请 人：刘浏 教授

报告摘要：In 1971-1973, Enflo constructed the example of a separable Banach space which fails the approximation property, and Johnson, Rosenthal, Zippin and Pełczyński independently obtained that a separable Banach space has the bounded approximation property (BAP) if and only if it is isomorphic to a complemented subspace of a Banach space with a Schauder basis. In 1999-2000, by the dilation technique for countably atomic case, Casazza, Han and Larson introduced the concept of frames for Banach spaces, and proved that it exists if and only if the space has the BAP. Recently, we solved the reflexivity problem for Schauder frames and atomic decompositions, and systematically developed the general Banach dilation theory from commutative to non-commutative operator-valued measures and strongly continuous operators (Memoirs of the AMS). Moreover, We also proved the operator space version of the above Pełczyński, Casazza, Han and Larson's theorems, then constructed the concrete frame example for reduced free group C*-algebra.

In the past twenty years, the interests on nonlinear theory for Banach spaces keep increasing, including ball covering property, Mazur-Ulam extension on spheres, (near-)isometric and (coarse) Lipschitz embedding theory, and also approximation properties involving Lipschitz functions and the Lipschitz free spaces. The famous Godefroy-Kalton theorem says that the Lipchitz BAP and the BAP are equivalent for every Banach space, and Godefroy and Ozawa characterized the metric spaces whose free space has the BAP. By our nonlinear Banach dilation technique, we generalize the above Godefroy-Kalton equivalence theorem to wider case on operators and frames for Banach spaces.

报告人简介：刘锐，博士，南开大学数学科学学院教授，博士生导师，主要从事泛函分析与相关领域研究，本科毕业于南开大学陈省身数学基地班（Chern Class），公派Texas A&M大学联合培养，导师为Banach空间方向著名数学家美国数学会会士Thomas Schlumprecht，曾获由法国科学院院士Gilles Pisier提供的TAMU Owen讲席基金资助，至今多篇研究论文发表在J. Funct. Anal., Memoirs Amer. Math. Soc., Fundamenta Mathematicae, J. Fourier Anal. Appl., Studia Math., 中国科学等国内外数学期刊，入选南开大学百名青年科学带头人计划与天津市131创新人才计划，先后主持国家自然科学基金面上项目2项和青年基金1项，入选全国泛函分析空间理论联络组与现代分析数学及其应用学术委员会。