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 sys...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 (Ⅱ<sub>1</sub>(⋅,⋅), π<sub>1</sub>(⋅)) 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 (Ⅱ<sub>1</sub>(⋅,⋅), π<sub>1</sub>(⋅)) if and only if V is finitely representable in {M<sub>n</sub>}<sub>n∈N</sub> in the system (Ⅱ<sub>1</sub>(⋅,⋅), π<sub>1</sub>(⋅)). At last, we show that an operator space V is finitely representable in {M<sub>n</sub>}<sub>n∈N</sub> in the system (Ⅱ<sub>1</sub>(⋅,⋅), π<sub>1</sub>(⋅)) if and only if V = C.展开更多
This study aims to introduce the notions of injectivity, local reflexivity, exactness, and nuclearity in the system(Γ2c(·, ·), γ2c(·)). We find that every dual operator space is injective in the syst...This study aims to introduce the notions of injectivity, local reflexivity, exactness, and nuclearity in the system(Γ2c(·, ·), γ2c(·)). We find that every dual operator space is injective in the system(Γ2c(·, ·), γ2c(·)) and nuclearity is equivalent to exactness in this system. As a corollary, we prove that Kirchberg’s conjecture on the equivalence of exactness and local reflexivity for C*-algebras is false in this system, i.e., there exists a C*-algebra A that is locally reflexive in this system but is not exact in this system.展开更多
文摘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 (Ⅱ<sub>1</sub>(⋅,⋅), π<sub>1</sub>(⋅)) 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 (Ⅱ<sub>1</sub>(⋅,⋅), π<sub>1</sub>(⋅)) if and only if V is finitely representable in {M<sub>n</sub>}<sub>n∈N</sub> in the system (Ⅱ<sub>1</sub>(⋅,⋅), π<sub>1</sub>(⋅)). At last, we show that an operator space V is finitely representable in {M<sub>n</sub>}<sub>n∈N</sub> in the system (Ⅱ<sub>1</sub>(⋅,⋅), π<sub>1</sub>(⋅)) if and only if V = C.
基金National Natural Science Foundation of China (Grant No. 11871423)。
文摘This study aims to introduce the notions of injectivity, local reflexivity, exactness, and nuclearity in the system(Γ2c(·, ·), γ2c(·)). We find that every dual operator space is injective in the system(Γ2c(·, ·), γ2c(·)) and nuclearity is equivalent to exactness in this system. As a corollary, we prove that Kirchberg’s conjecture on the equivalence of exactness and local reflexivity for C*-algebras is false in this system, i.e., there exists a C*-algebra A that is locally reflexive in this system but is not exact in this system.
