Vector Lattice Powers: f-Algebras and Functional Calculus

Abstract

Let s ∈ {2.3,…} and E be an Archimedean vector lattice. We prove that there exists a unique pair (E Ⓢ ,Ⓢ), where E Ⓢ is an Archimedean vector lattice and Ⓢ:E× ··· ×E (s times) → E Ⓢ is a symmetric lattice s-morphism, such that for every Archimedean vector lattice F and every symmetric lattice s-morphism T:E × ··· × E (s times) → F, there exists a unique lattice homomorphism T Ⓢ :E Ⓢ → F such that T = T Ⓢ ○Ⓢ. We give two approaches to construct (E Ⓢ ,Ⓢ) based on f-algebras and functional calculus, respectively, provided that E is also uniformly complete.