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nuclearity and Finite-Representability in the System of Completely Integral Mapping Spaces

Acta Mathematica Sinica,English Series 2024年第5期40卷 1197-1214页

作者： Zhe DONG Ji Cheng TAO Ya Fei ZHAO Department of Mathematics Zhejiang UniversityHangzhou 310027P.R.China Department of Mathematics China Jiliang UniversityHangzhou 310018P.R.China Department of Mathematics Zhejiang International Studies UniversityHangzhou 310023P.R.China

In this paper,we investigate local properties in the system of completely integral mapping spaces.We introduce notions of injectivity,local reflexivity,exactness,nuclearity,finite-represent ability and WEP in the system of completely integral mapping spaces.First we obtain that any finite-dimensional operator space is injective in the system of completely integral mapping spaces.Furthermore we prove that C is the unique nuclear operator space and the unique exact operator space in this system.We also show that C is the unique operator space which is finitely representable in{T_(n)}n∈Nin this system.As corollaries,Kirchberg’s conjecture and QWEP conjecture in the system of completely integral mapping spaces are false.

关键词： Completely integral mapping space injectivity exactness nuclearity finite-representability WEP

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Kirchberg’s conjecture in the system of Hilbert space factorable mapping spaces

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Science China Mathematics 2020年第6期63卷 1125-1138页

作者： Zhe Dong School of Mathematical Sciences Zhejiang UniversityHangzhou 310027China

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.

关键词： Hilbert space factorable mapping space injectivity local reflexivity exactness nuclearity

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The Local Theory of Completely 1-Summing Mapping Spaces

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Journal of Applied Mathematics and Physics 2023年第6期11卷 1570-1579页

作者： Yafei Zhao Yuanyi Wang Department of Mathematics Zhejiang International Studies University Hangzhou China College of Science and Technology Ningbo University Ningbo China

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 {Mn}n∈N in the system (Ⅱ1(⋅,⋅), π1(⋅)). At last, we show that an operator space V is finitely representable in {Mn}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

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Topological Influence of the Presence of Fully-λ-Bases in Locally Convex Spaces

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Acta Mathematica Sinica,English Series 1996年第1期12卷 105-112页

作者： Deheri G. M. Department of Mathematics Sardar Patel University VALLABH VIDYANAGAR-388120 Via Anand (Gujarat-INDIA) Sardar Patel University 印度

This article reveals the topological impact of fully-λ-bases in locally convex spaces where λ carries either the traditional normal topology or the fairly generalized σμ-topology of Ruckle. It has been established that the generalized nuclearity of λ plays a significan role in influencing the topology of the space. Further, the equivalence of normal topology and the topology arising out of the fully-λ-base (λ being equipped with normal topology or σμ-topology) has been investigated.

关键词： Locally convex space Fully-λ-base nuclearity

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