A complete classification of the spaces of compact operators on $C([1,\alpha ], l_{p})$ spaces, $1<p< \infty$

Abstract:

We complete the classification, up to isomorphism, of the spaces of compact operators on $C([1, \gamma ], l_{p})$ spaces, $1<p< \infty$. In order to do this, we classify, up to isomorphism, the spaces of compact operators ${\mathcal K}(E, F)$, where $E= C([1, \lambda ], l_{p})$ and $F=C([1, \xi ], l_q)$ for arbitrary ordinals $\lambda$ and $\xi$ and $1< p \leq q< \infty$.

More precisely, we prove that it is relatively consistent with ZFC that for any infinite ordinals $\lambda$, $\mu$, $\xi$ and $\eta$ the following statements are equivalent:

1. [(a)] ${\mathcal K}(C([1, \lambda ], l_{p}), C([1, \xi ], l_{q}))$ is isomorphic to ${\mathcal K}(C([1, \mu ], l_{p}), C([1, \eta ], l_{q})) .$

2. [(b)] $\lambda$ and $\mu$ have the same cardinality and $C([1,\xi ])$ is isomorphic to $C([1, \eta ])$ or there exists an uncountable regular ordinal $\alpha$ and $1 \leq m, n < \omega$ such that $C([1, \xi ])$ is isomorphic to $C([1, \alpha m])$ and $C([1, \eta ])$ is isomorphic to $C([1, \alpha n])$.

Moreover, in ZFC, if $\lambda$ and $\mu$ are finite ordinals and $\xi$ and $\eta$ are infinite ordinals, then the statements (a) and (b$’$) are equivalent.

1. [(b$’$)] $C([1,\xi ])$ is isomorphic to $C([1, \eta ])$ or there exists an uncountable regular ordinal $\alpha$ and $1 \leq m, n \le \omega$ such that $C([1, \xi ])$ is isomorphic to $C([1, \alpha m])$ and $C([1, \eta ])$ is isomorphic to $C([1, \alpha n])$.