Local properties of ternary rings of operators and their linking \(C^{*}\)-algebras
A ternary ring of operators between Hilbert spaces \(K\) and \(H\) is a norm closed subspace \(V\) of \(B(K, H)\), which is closed under the triple product \[ (x,y,z)\in V\times V\times V\mapsto xy^* z\in V. \] In the paper under review, it is proved that some local properties, such as nuclearity, exactness, and local reflexivity of ternary ring of operators are closely related to the local properties of their linking \(C^*\)-algebras. Haagerup's decomposition property for completely bounded maps and Pisier's \(\delta\)-norm are also naturally generalized to a ternary ring of operators. In the Appendix, it is shown that for each \(n> 2\), there exists an operator space which is \((n+1)\)-locally reflexive, and is only \(\lambda\)-locally reflexive for \(\lambda\geq{n\over\sqrt{n-1}}\).
scientific article; zbMATH DE number 1888358
scientific article