On a problem of H.\,P.\ Rosenthal concerning operators on \(C\)[0,1] (Q932169)

Let \(K\) be a compact Hausdorff space. If \((X,\|\cdot\|)\) is a Banach space with norm \(\|\cdot\|\), let \(B_X\) denote the unit ball of \(X\), so that \(B_X= \{x\in X:\| x\|\leq 1\}\), let \(X^*\) denote the dual space of bounded linear functionals on \(X\), and let \(M\) denote the \(w^*\)-compact subset of \(B_{C(K)^*}\) which is not separable in the \(C(K)^*\)-norm. In the main theorem of this paper, the author states that if \(K\) is uncountable, compact and metrizable, if \(X\) is a closed linear infinite-dimensional subspace of \(C(K)\), and if \(M\subset B_{C(K)^*}\) is \(w^*\)-compact and isometric with \(X\), then \(M\) is not separable in the \(C(K)^*\)-norm. The main statement is indicated as being equivalent to an affirmative confirmation of a problem of \textit{H.\,P.\thinspace Rosenthal} [Isr.\ J.\ Math.\ 13, 361--378 (1973; Zbl 0253.46048)], which enquires about the existence of a linear subspace \(Y\) of \(C(K)\) isomorphic with \(C(K)\).