Some results on the lattice of closed ideals of $\mathcal L^r(X)$ for $X$ of the form $(\bigoplus_i \ell _p^i)_q$
Volume 261 / 2021
Studia Mathematica 261 (2021), 25-53
MSC: Primary 46H10, 47L10; Secondary 46B42, 47B65.
DOI: 10.4064/sm200305-25-1
Published online: 31 May 2021
Abstract
We study the lattice of closed (order and algebra) ideals of $\mathcal L^r(X)$ when $X$ is a Banach lattice of the form $(\bigoplus _i \ell _p^i)_q$ $(p\in [1,\infty ]$, $q\in [1,\infty )\cup \{0\} \,\&\, p\ne q)$. We show that for every such $X$, $\mathcal L^r(X)$ has a unique maximal (order and algebra) ideal. For $1 \lt p \lt \infty $ and $q\in \{0,1\}$, we show, in particular, that the lattice of closed (order and algebra) ideals of $\mathcal L^r(X)$ contains at least five distinct ideals.
