A new complemented subspace for the Lorentz sequence spaces, with an application to its lattice of closed ideals

Abstract

We show that every Lorentz sequence space (w, ) admits a 1-complemented subspace distinct from ℓ and containing no isomorph of (w, ). In the general case, this is only the second nontrivial complemented subspace in (w, ) yet known. We also give an explicit representation of in the special case w = ( − ) ∞ =1 (0 < < 1) as the ℓ -sum of finite-dimensional copies of (w, ). As an application, we find a sixth distinct element in the lattice of closed ideals of L ( (w, )), of which only five were previously known in the general case.