CLOSED IDEALS OF OPERATORS ON AND COMPLEMENTED SUBSPACES OF BANACH SPACES OF FUNCTIONS WITH COUNTABLE SUPPORT Academic Article

Let lambda be an infinite cardinal number and let c ( ) 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 c ( ) ell _infty ^c(lambda ) , proving that they are isomorphic to c ( ) ell _infty ^c(kappa ) for some cardinal number kappa . Then we show that the Banach algebra of all bounded linear operators on c ( ) 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 B ( X ) mathscr {B}(X) , where X = c