Instytut Matematyczny Polskiej Akademii Nauk / pl / Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

Streszczenie

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.