Extension of Finite Rank Operators and Local Structures in Operator Ideals

Abstract

We develop general techniques and present an approach to solve the problem of constructing a maximal Banach ideal $({\frak A},{\bf A)}$ which does not satisfy a transfer of the norm estimation in the principle of local reflexivity to its norm ${\bf A}$. This approach leads us to the investigation of product operator ideals containing ${\frak L}_2$ (the collection of all Hilbertian operators) as a factor. Using the local properties of such operator ideals -- which are typical examples of ideals with property (I) and property (S) --, trace duality and an extension of suitable finite rank operators even enable us to show that ${\frak L}_\infty $ cannot be totally accessible -- answering an open question of Defant and Floret.