Boolean-Valued Analysis of Order-Bounded Operators

Abstract

This is a survey of some recent applications of Boolean valued models of set theory to order bounded operators in vector lattices.Everywhere below B denotes a complete Boolean algebra, while V (B) stands for the corresponding Boolean valued universe (the universe of B-valued sets). A partition of unity in B is a family (b ξ ) ξ∈Ξ ⊂ B with ξ∈Ξ b ξ = 1 and b ξ ∧b η = 0 for ξ = η.By a vector lattice throughout the sequel we will mean a real Archimedean vector lattice, unless specified otherwise. We let := denote the assignment by definition, while N, Q, R, and C symbolize the naturals, the rationals, the reals, and the complexes. We denote the Boolean algebras of bands and band projections in a vector lattice X by B(X) and P(X); and we let X u stand for the universal completion of a vector lattice X.The ideal center Z (X) of a vector lattice X is an f -algebra. Let Orth(X) and Orth ∞ (X) stand for the f -algebras of orthomorphisms and extended orthomorphisms, respectively X. Then Z (X) ⊂ Orth(X) ⊂ Orth ∞ (X). The space of all order bounded linear operators from X to Y is denoted by L ∼ (X, Y ). The Riesz-Kantorovich Theorem tells us that if Y is a Dedekind complete vector lattice then so is L ∼ (X, Y ). Escher RulesNow, we present a remarkable interplay between V and V (B) which is based on the operations of canonical embedding, descent, and ascent.1.2.1. We start with the canonical embedding of the von Neumann universe into the Boolean valued universe. Given x ∈ V, we denote by x ∧ the standard name of x in V (B) ; i.e., the element defined by the following recursion schema:∅ ∧ := ∅, dom(x ∧ ) := {y ∧ : y ∈ x}, im(x ∧ ) := {1}.