主题：Approximation properties, frames, and dilations of operator-valued measures
刘锐，南开大学数学科学学院副教授，博士生导师，南开大学百名青年学科带头人。 主要研究方向：泛函分析，空间理论及其应用。南开大学与美国德州农工大学联合培养博士。曾受邀访问了美国德州大学奥斯汀分校、伊利诺伊大学香槟分校、德州农工大学、中佛罗里达大学等名校。 近年来在Memoirs of American Mathematical Society，Journal of Functional Analysis，Fundamenta Mathematicae，Journal of Fourier Analysis and Applications等国际顶尖期刊上发表了多篇学术论文。主持完成了国家自然科学基金青年项目，正主持国家自然科学基金面上项目。
This talk is on the intersection topics between functional analysis and applied harmonic analysis: 1) We introduce the concept of (Schauder) frames for Banach and operator spaces, show the connection with the bounded approximation property and complemented embedding, and give the duality theorems for frames and associated basis in reflexive Banach spaces. We give a concrete frame for reduced free group C*-algebra, and also prove that a separable operator space has the completely bounded approximation property if and only if it has a completely bounded frame if and only if it is completely isomorphic to a completely complemented subspace of an operator space with a completely bounded basis. 2) This is on a general dilation theory of operator-valued measures and frames for Banach spaces, motivated by the observation that there is a connection between the analysis of dual pairs of frames (both the discrete and the continuous theory) and the dilation theory of operator-valued measures on Banach spaces. As a continuation of our recent work, we show that every operator-valued system of imprimitivity with a projective isometric group representation has dilation to a spectral system of imprimitivity acting on a larger Banach space, and also prove that every operator-valued measure with bounded p-variation can be dilated to a projection-valued measure with the same variation property on a larger Banach space.