  Let $\lambda$ be an infinite cardinal number and let $\ell_\infty^c(\lambda)$
denote the subspace of $\ell_\infty(\lambda)$ consisting of all functions that
assume at most countably many non-zero values. We classify all infinite
dimensional complemented subspaces of $\ell_\infty^c(\lambda)$, proving that
they are isomorphic to $\ell_\infty^c(\kappa)$ for some cardinal number
$\kappa$. Then we show that the Banach algebra of all bounded linear operators
on $\ell_\infty^c(\lambda)$ or $\ell_\infty(\lambda)$ 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 $\mathscr{B}(X)$, where $X =
c_0(\lambda)$ or $X=\ell_p(\lambda)$ for some $p\in [1,\infty)$, and we
classify the closed ideals of $\mathscr{B}(\ell_\infty^c(\lambda))$ that
contains the ideal of weakly compact operators.
