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

1988年，阮志坚提出算子空间的概念，为算子代数和泛函分析的研究提供了新的视角。我们可以将算子空间结构引入算子空间的对偶空间以及算子空间上的所有完全正映射。将规范空间中的参数转移到运算符空间中的参数并不难。正如口号所说，“算子空间理论是泛函分析的量化”。I和M.NGAISA证明了算子空间背后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式的原始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第二次量化”。此外，我们发现了对合范数空间到具体数值半径算子空间的保留表示，而阮的表示根本不保留对合性。由此，我们找到了算子数值半径的Anto型定理的简单证明，该定理利用希尔伯特空间上有界算子的因式分解来描述数值半径。

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.