Complementation in the Fremlin vector lattice symmetric tensor products-II

Abstract

For a vector lattice E and $n \in \mathbb {N}$, let ${\bar{\otimes }}_{n,s}E$ denote the n-fold Fremlin vector lattice symmetric tensor product of E. For $m, n \in \mathbb {N}$ with $m > n$, we prove that (i) if ${\bar{\otimes }}_{m,s}E$ is uniformly complete then ${\bar{\otimes }}_{n,s}E$ is positively isomorphic to a complemented subspace of ${\bar{\otimes }}_{m,s}E$, and (ii) if there exists such that $\ker (\phi )$ is a projection band in E then ${\bar{\otimes }}_{n,s}E$ is lattice isomorphic to a projection band of ${\bar{\otimes }}_{m,s}E$. We also obtain analogous results for the n-fold Fremlin projective symmetric tensor product ${\hat{\otimes }}_{n,s,|\pi |}E$ of E where E is a Banach lattice.