Proceedings of the American Mathematical Society

Abstract:

We extend and provide a vector-valued version of some results of C. Samuel about the geometric relations between the spaces of nuclear operators ${\mathcal N}(E, F)$ and spaces of compact operators ${\mathcal K}(E, F)$, where $E$ and $F$ are Banach spaces $C(K)$ of all continuous functions defined on the countable compact metric spaces $K$ equipped with the supremum norm.

First we continue Samuel’s work by proving that ${\mathcal N} (C(K_{1}), C(K_{2}))$ contains no subspace isomorphic to ${\mathcal K} (C(K_{3}), C(K_{4}))$ whenever $K_1$, $K_{2}$, $K_{3}$ and $K_{4}$ are arbitrary infinite countable compact metric spaces.

Then we show that it is relatively consistent with ZFC that the above result and the main results of Samuel can be extended to $C(K_{1}, X)$, $C(K_{2}, Y)$, $C(K_{3}, X)$ and $C(K_{4}, Y)$ spaces, where $K_1$, $K_{2}$, $K_{3}$ and $K_{4}$ are arbitrary infinite totally ordered compact spaces; $X$ comprises certain Banach spaces such that $X^*$ are isomorphic to subspaces of $l_1$; and $Y$ comprises arbitrary subspaces of $l_p$, with $1<p< \infty$.

Our results cover the cases of some non-classical Banach spaces $X$ constructed by Alspach, by Alspach and Benyamini, by Benyamini and Lindenstrauss, by Bourgain and Delbaen and also by Argyros and Haydon.