The subprojectivity of the projective tensor product of two \(C(K)\) spaces with \(| K|=\aleph_{0}\) (Q2796731)

In 1964, \textit{R. J. Whitley} [Trans. Am. Math. Soc. 113, 252--261 (1964; Zbl 0124.06603)] introduced the concept of subprojective Banach spaces in order to study strictly singular operators. A Banach space \(X\) is said to be subprojective if every infinite-dimensional subspace of \(X\) contains a further infinite-dimensional subspace which is complemented in \(X\).NEWLINENEWLINERecently, \textit{T. Oikhberg} and \textit{E. Spinu} [J. Math. Anal. Appl. 424, No. 1, 613-635 (2015; Zbl 1338.46013)] made an extensive study of subprojective spaces. Suppose that \(K\) is a compact metrizable space. One result of Oikhberg and Spinu is that the projective tensor product \(C(K) \hat{\otimes} c_0\) is subprojective if and only if \(K\) is scattered. (Recall that a compact space \(K\) is countable if and only if \(K\) is metrizable and scattered.)NEWLINENEWLINEThe authors show that if \(K_1\) and \(K_2\) are two compact metric spaces, then \(C(K_1) \hat{\otimes} C(K_2)\) is subprojective if and only if \(K_1\) and \(K_2\) are countable. The proof uses the other main result of the paper which says that if \(K_1\) and \(K_2\) are two countable compact metric spaces, then \(C(K_1) \hat{\otimes} C(K_2)\) is \(c_0\)-saturated.