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 $(\mathfrak{A},\mathbf{A})$ which does not satisfy a transfer of the norm estimation in the principle of local reflexivity to its norm $\mathbf{A}$. This approach leads us to the investigation of product operator ideals containing ${\mathfrak{L}}_2$ (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 ${\mathfrak{L}}_{\infty }$ cannot be totally accessible -- answering an open question of Defant and Floret.

LaTeX 2e, 26 pages