The Local Theory of Completely 1-Summing Mapping Spaces  

作　　者：Yafei Zhao Yuanyi Wang Yafei Zhao;Yuanyi Wang(Department of Mathematics, Zhejiang International Studies University, Hangzhou, China;College of Science and Technology, Ningbo University, Ningbo, China)

机构地区：[1]Department of Mathematics, Zhejiang International Studies University, Hangzhou, China [2]College of Science and Technology, Ningbo University, Ningbo, China

出　　处：《Journal of Applied Mathematics and Physics》2023年第6期1570-1579,共10页应用数学与应用物理（英文）

摘　　要：In this paper, we investigate local properties in the system of completely 1-summing mapping spaces. We introduce notions of injectivity, local reflexivity, exactness, nuclearity and finite-representability in the system of completely 1-summing mapping spaces. First we obtain that if V has WEP, V is locally reflexive in the system (Ⅱ₁(⋅,⋅), π₁(⋅)) if and only if it is locally reflexive in the system (Ⅰ(⋅,⋅), t(⋅)). Furthermore we prove that an operator space V ⊆ B(H) is exact in the system (Ⅱ₁(⋅,⋅), π₁(⋅)) if and only if V is finitely representable in {M_n}_n∈N in the system (Ⅱ₁(⋅,⋅), π₁(⋅)). At last, we show that an operator space V is finitely representable in {M_n}_n∈N in the system (Ⅱ₁(⋅,⋅), π₁(⋅)) if and only if V = C.In this paper, we investigate local properties in the system of completely 1-summing mapping spaces. We introduce notions of injectivity, local reflexivity, exactness, nuclearity and finite-representability in the system of completely 1-summing mapping spaces. First we obtain that if V has WEP, V is locally reflexive in the system (Ⅱ₁(⋅,⋅), π₁(⋅)) if and only if it is locally reflexive in the system (Ⅰ(⋅,⋅), t(⋅)). Furthermore we prove that an operator space V ⊆ B(H) is exact in the system (Ⅱ₁(⋅,⋅), π₁(⋅)) if and only if V is finitely representable in {M_n}_n∈N in the system (Ⅱ₁(⋅,⋅), π₁(⋅)). At last, we show that an operator space V is finitely representable in {M_n}_n∈N in the system (Ⅱ₁(⋅,⋅), π₁(⋅)) if and only if V = C.

关 键 词：Completely 1-Summing Mapping Space Injectivity NUCLEARITY Local Reflexivity EXACTNESS Finite-Representability and Operator Space 

分 类 号：O17[理学—数学]