Spaces of nuclear and compact operators without a complemented copy of $C(\omega ^{\omega })$
Résumé
$X\sim Y$ denotes that $X$ and $Y$ are linearly isomorphic Banach spaces. Let $\omega, \omega_1$ denote the first infinite and the first uncountable ordinal, respectively. Let $\omega\leq\alpha\leq\beta<\omega_1$, and let $\eta, \xi<\omega_1$ with $\bar\eta=\bar\xi$. The topic of the paper under review is to find conditions on the Banach spaces $X,Y,Z,W$ involved such that either of the two statements \par \noindent $\mathcal{N}(X\oplus C(\xi),Y\oplus C(\alpha))\sim \mathcal{N}(X\oplus C(\eta),Y\oplus C(\beta))$, \newline $\mathcal{K}(X\oplus C(\xi),Y\oplus C(\alpha))\sim \mathcal{K}(X\oplus C(\eta),Y\oplus C(\beta))$ \par \noindent is equivalent to $\beta<\alpha^\omega$. The solution of these two problems gives two generalizations of the classical Bessaga-Pelczynski result saying that, for $\omega\leq\alpha\leq\beta<\omega_1$, $C(\alpha)\sim C(\beta)$ if and only if $\beta<\alpha^\omega$. At the same time, one obtains generalizations of the following result of the second named author: For $\omega\leq\alpha\leq\beta<\omega_1$, each of the two statements \par \noindent $\mathcal{N}(C(\alpha))\sim \mathcal{N}(C(\beta))$,\newline $\mathcal{K}(C(\alpha))\sim \mathcal{K}(C(\beta))$ \par \noindent is equivalent to $\beta<\alpha^\omega$, see [{\it C. Samuel}, Proc. Am. Math. Soc. 137, No. 3, 965--970 (2009; Zbl 1172.46009)].