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Let $\lambda$ be an infinite cardinal number and let $\ell_\infty^c(\lambda)$denote the subspace of $\ell_\infty(\lambda)$ consisting of all functions thatassume at most countably many non-zero values. We classify all infinitedimensional complemented subspaces of $\ell_\infty^c(\lambda)$, proving thatthey are isomorphic to $\ell_\infty^c(\kappa)$ for some cardinal number$\kappa$. Then we show that the Banach algebra of all bounded linear operatorson $\ell_\infty^c(\lambda)$ or $\ell_\infty(\lambda)$ has the unique maximalideal consisting of operators through which the identity operator does notfactor. Using similar techniques, we obtain an alternative to Daws' approachdescription of the lattice of all closed ideals of $\mathscr{B}(X)$, where $X =c_0(\lambda)$ or $X=\ell_p(\lambda)$ for some $p\in [1,\infty)$, and weclassify the closed ideals of $\mathscr{B}(\ell_\infty^c(\lambda))$ thatcontains the ideal of weakly compact operators.

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