Abstract
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-representability 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 {Tn}n∈ℕ in this system. As corollaries, Kirchberg’s conjecture and QWEP conjecture in the system of completely integral mapping spaces are false.
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The authors give their sincere thanks to the referees for their valuable comments and suggestions.
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Supported by the National Natural Science Foundation of China (Grant No. 11871423) and Zhejiang Provincial Natural Science Foundation of China (Grant No. LQ21A010015)
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Dong, Z., Tao, J.C. & Zhao, Y.F. Nuclearity and Finite-Representability in the System of Completely Integral Mapping Spaces. Acta. Math. Sin.-English Ser. 40, 1197–1214 (2024). https://doi.org/10.1007/s10114-023-2103-0
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DOI: https://doi.org/10.1007/s10114-023-2103-0