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

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 {T_n}_n∈ℕ in this system. As corollaries, Kirchberg’s conjecture and QWEP conjecture in the system of completely integral mapping spaces are false.