Closed ideals of operators on and complemented subspaces of Banach spaces of functions with countable support

Let $\lambda$ be an infinite cardinal number and let ${\mathrm{\ell }}_{\mathrm{\infty }}^c(\lambda )$ denote the subspace of ${\mathrm{\ell }}_{\mathrm{\infty }}(\lambda )$ consisting of all functions that assume at most countably many non-zero values. We classify all infinite dimensional complemented subspaces of ${\mathrm{\ell }}_{\mathrm{\infty }}^c(\lambda )$, proving that they are isomorphic to ${\mathrm{\ell }}_{\mathrm{\infty }}^c(\kappa )$ for some cardinal number $\kappa$. Then we show that the Banach algebra of all bounded linear operators on ${\mathrm{\ell }}_{\mathrm{\infty }}^c(\lambda )$ or ${\mathrm{\ell }}_{\mathrm{\infty }}(\lambda )$ has the unique maximal ideal consisting of operators through which the identity operator does not factor. Using similar techniques, we obtain an alternative to Daws’ approach description of the lattice of all closed ideals of $\mathcal{B}(X)$, where $X = c_0(\lambda)$or$X=\ell_p(\lambda)$forsome$p\in [1,\infty)$, and we classify the closed ideals of $\mathcal{B}({\mathrm{\ell }}_{\mathrm{\infty }}^c(\lambda ))$ that contains the ideal of weakly compact operators.