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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. [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions for nuclearity of Cuntz–Nica–Pimsner algebras for a variety of quasi-lattice ordered groups. First we deal with the free abelian lattice case. We use this as a stepping-stone to tackle product systems over quasi-lattices that are controlled by the free abelian lattice and satisfy a minimality property. Our setting accommodates examples like the Baumslag–Solitar lattice for $n=m>0$ and the right-angled Artin groups. More generally, the class of quasi-lattices for which our results apply is closed under taking semi-direct and graph products. In the process we accomplish more. Our arguments tackle Nica–Pimsner algebras that admit a faithful conditional expectation on a small fixed point algebra and a faithful copy of the coefficient algebra. This is the case for CNP-relative quotients in-between the Toeplitz–Nica–Pimsner algebra and the Cuntz–Nica–Pimsner algebra. We complete this study with the relevant results on exactness. [ABSTRACT FROM AUTHOR]

This study aims to introduce the notions of injectivity, local reflexivity, exactness, and nuclearity in the system ( (Γ 2 c (⋅ , ⋅) , γ 2 c (⋅)) ). We find that every dual operator space is injective in the system ( (Γ 2 c (⋅ , ⋅) , γ 2 c (⋅)) ) 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. [ABSTRACT FROM AUTHOR]

Kirchberg's Embedding Problem (KEP) asks whether every separable C ⁎ algebra embeds into an ultrapower of the Cuntz algebra O 2 . In this paper, we use model theory to show that this conjecture is equivalent to a local approximate nuclearity condition that we call the existence of good nuclear witnesses . In order to prove this result, we study general properties of existentially closed C ⁎ algebras. Along the way, we establish a connection between existentially closed C ⁎ algebras, the weak expectation property of Lance, and the local lifting property of Kirchberg. The paper concludes with a discussion of the model theory of O 2 . Several results in this last section are proven using some technical results concerning tubular embeddings, a notion first introduced by Jung for studying embeddings of tracial von Neumann algebras into the ultrapower of the hyperfinite II 1 factor. [ABSTRACT FROM AUTHOR]

Abstract: We continue our study of tensor products in the operator system category. We define operator system quotients and exactness in this setting and refine the notion of nuclearity by studying operator systems that preserve various pairs of tensor products. One of our main goals is to relate these refinements of nuclearity to the Kirchberg conjecture. In particular, we prove that the Kirchberg conjecture is equivalent to the statement that every operator system that is (min,er)-nuclear is also (el,c)-nuclear. We show that operator system quotients are not always equal to the corresponding operator space quotients and then study exactness of various operator system tensor products for the operator system quotient. We prove that an operator system is exact for the min tensor product if and only if it is (min,el)-nuclear. We give many characterizations of operator systems that are (min,er)-nuclear, (el,c)-nuclear, (min,el)-nuclear and (el,max)-nuclear. These characterizations involve operator system analogues of various properties from the theory of C*-algebras and operator spaces, including the WEP and LLP. [Copyright &y& Elsevier]