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 $d(\text{w},p)$ admits a 1-complemented subspace $Y$ distinct from ${\ell }_p$ and containing no isomorph of $d(\text{w},p)$. In the general case, this is only the second nontrivial complemented subspace in $d(\text{w},p)$ yet known. We also give an explicit representation of $Y$ in the special case $\text{w}=(n^{-\theta }{)}_{n=1}^{\infty }$ ($0<\theta <1$) as the ${\ell }_p$-sum of finite-dimensional copies of $d(\text{w},p)$. As an application, we find a sixth distinct element in the lattice of closed ideals of $\mathcal{L}(d(\text{w},p))$, of which only five were previously known in the general case.