Spaces of nuclear and compact operators without a complemented copy of \(C(\omega ^{\omega })\) (Q1941091)

\(X\sim Y\) 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 \noindent \(\mathcal{N}(X\oplus C(\xi),Y\oplus C(\alpha))\sim \mathcal{N}(X\oplus C(\eta),Y\oplus C(\beta))\), \newline \(\mathcal{K}(X\oplus C(\xi),Y\oplus C(\alpha))\sim \mathcal{K}(X\oplus C(\eta),Y\oplus C(\beta))\) \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 \noindent \(\mathcal{N}(C(\alpha))\sim \mathcal{N}(C(\beta))\),\newline \(\mathcal{K}(C(\alpha))\sim \mathcal{K}(C(\beta))\) \noindent is equivalent to \(\beta<\alpha^\omega\), see [\textit{C. Samuel}, Proc. Am. Math. Soc. 137, No. 3, 965--970 (2009; Zbl 1172.46009)].