空间理论中与框架相关的逼近性质和扩张理论的研究

Approximation properties and dilation problems are fundamental research topics in space theory and related to each other, the project research content is mainly from recent research work of Casazza, Junge, Han, Larson, Ruan and Xu, mainly use frames as analysis bridge and tool to deeply research approximation properties and dilation theory on operator spaces, Banach spaces, non-commutative L_p spaces (including Fourier algebra) and reduced group C*-algebras, includes: research Hilbert operator space the Oikhberg-Ricard example to obtain local property characterization to be completely isomorphic to a subspace of operator spaces with completely bounded base; research on constructive and existence properties problems of completely bounded base and frames of non-commutative L_p spaces (including Fourier algebra) and reduced C*-algebras on weakly amenable groups (such as free group) associated with natural operator space structures; prove the duality theorem that unconditional frames on reflexive Banach spaces have reflexive associated spaces with unconditional bases; moreover, based on these, this project will systematically research the duality dilation theory on operator-valued measures on Banach spaces and imprimitivity systems with (semi-)group actions in abstract harmonic analysis.

逼近性质和扩张问题是空间理论中相互关联的基础研究方向，本项目研究内容主要源自Casazza、Junge、Han、Larson、Ruan和Xu近些年的研究工作，主要针对算子空间、Banach空间、非交换L_p空间(包括Fourier代数)和约化群C*-代数用框架为工具研究逼近性质和对偶扩张问题，具体包括：研究Oikhberg和Ricard的Hilbert型算子空间的例子，进而得到完全同构嵌入有完全有界基的算子空间的局部刻画性质；在自然的算子空间结构下，考虑弱顺从群(比如自由群)上非交换L_p空间(包括Fourier代数)和约化C*-代数中完全有界基和框架的构造性及存在性问题；证明自反Banach空间上无条件框架存在自反伴随空间及无条件基的对偶定理；并且，以此为基础，本项目将系统研究空间上算子值测度和抽象调和分析中(半)群作用下的Imprimitivity系统的对偶扩张理论。

逼近性质和扩张问题是空间理论中相互关联的基础研究方向，本项目研究内容主要源自泛函分析与框架理论的交叉研究领域在近些年的研究工作，主要针对算子空间和Banach空间为工具研究逼近性质和扩张问题，具体研究成果包括：证明了交换情况下算子值测度的有界p-变差版本的扩张定理；得到p-算子空间上的完全有界逼近性质和完全有界框架存在性的等价证明；给出代数同态扩张在有限维时的完整刻画；应用在由连续框架向量值积分生成的概率正算子值测度即量子信道的量子探测和单射性问题上并给出向量刻画。

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