Spaceability in sets of operators on \(C(K)\) (Q1935853)

Let \(C(K)\) be the Banach space, with respect to the supremum norm, of real-valued continuous functions on a Hausdorff compact topological space \(K\). An operator \(T\) on \(C(K)\) is a weak multiplication (resp., a weak multiplier) if there are \(g\in C(K)\) and a weakly compact operator \(S\) on \(C(K)\) such that \(T=g\mathrm{I}_d=+S\) (resp., if \(\lim T(e_n)(x_n)= 0\) for every pairwise disjoint bounded sequence \((e_n)\) in \(C(K)\) and for every sequence \((x_n)\) in \(K\) with \(e_n(x_n)= 0\) for all \(n\)); every weak multiplication is a weak multiplier. In the paper under review, the authors prove that, if there is an operator on \(C(K)\) which is not a weak multiplier, then the quotient of the space of all operators on \(C(K)\) by the subspace of all weak multipliers is an infinite-dimensional space. They also show that, if every operator on \(C(K)\) is a weak multiplication and \(L= K\times[0,1]\), then there exists an operator \(J\) on \(C(L)\) so that \(J\) is not a weak multiplier and every operator on \(C(L)\) is of the form \(gJ+ W\), where \(g\in C(L)\) and \(W\) is a weak multiplication.