A Study of Embeddings of Normed Spaces into Numerical Radius Operator Spaces

1988年阮志杰引入算子空间的概念，为研究算子代数和泛函分析提供了一个新的视角。我们可以在算子空间的对偶空间和算子空间上所有的完全正映射中引入算子空间结构。将赋范空间中的参数转换为算子空间中的参数并不难。正如一句流行语所说，“算符空间理论是泛函分析的量子化”。I和M.NGAISA证明了在算子空间后面存在各种数值半径算子空间。给定一个算子空间，我们可以找到与满足恒等的原范数相对应的各种数值半径范数。我们可以说“数值半径算子空间理论是泛函分析的二次量化”。此外，我们发现对合赋范空间的守恒表示为具体的数值半径算子空间，而阮氏表示根本不守恒对合。利用Hilbert空间上有界算子的因式分解来描述数值半径，得到了算子数值半径的Anto型定理的简单证明。

In 1988, the notion of operator spaces introduced by Z.J.Ruan shed light on a new point of view for studying not only on operator algebra but also on functional analysis. We can introduce an operator space structure into the dual space of operator spaces and all of completely positive maps on operator spaces. It is not hard to transfer the arguments in normed spaces into those in operator spaces. As the catch phrase, it is said that " The operator space theory is a quantization of the functional analysis".I and M.NGAISA proved the existence of various numerical radius operator spaces behind operator spaces. Given an operator space, we can find the various numerical radius norms which corresponds to the original norm satisfying an identity. We might say that " The numerical radius operator space theory is a second quantization of the functional analysis".Moreover we found an-preserving representation of the involutive normed space into a concrete numerical radius operator space, while the Ruan's representation did not preserve the involution at all. As the consequence, we found the simple proof of Anto type theorem for numerical radius of operators, which describe the numerical radius by using the factorization of bounded operators on Hilbert spaces.