Bidual extensions of Riesz multimorphisms

TL;DR

This paper investigates the properties of Arens and Aron-Berner extensions of Riesz multimorphisms, establishing conditions under which these extensions preserve the Riesz multimorphism structure in various Banach lattices.

Contribution

It proves that all Arens extensions of finite rank Riesz multimorphisms are Riesz multimorphisms and extends these results to certain Banach lattices for Aron-Berner extensions.

Findings

All Arens extensions of finite rank Riesz multimorphisms coincide and are Riesz multimorphisms.

For specific Banach lattices, all Aron-Berner extensions of F-valued Riesz multimorphisms are Riesz multimorphisms.

Partial results are obtained for arbitrary Riesz multimorphisms.

Abstract

We prove that all Arens extensions of finite rank Riesz multimorphisms taking values in Archimedean Riesz spaces coincide and are Riesz multimorphisms. Partial results for arbitrary Riesz multimorphisms are obtained. We also prove that, for a class of Banach lattices $F$, which includes $F = c_0, \ell_p, c_0(\ell_p), \ell_p(c_0), \ell_p(\ell_s), 1 < p,s < \infty$, among many others, all Aron-Berner extensions of $F$-valued Riesz multimorphisms between Banach lattices are Riesz multimorphisms.