The yosida space and representation of the projectable hull of an archimedean ℓ-group with weak unit

Abstract

W is the category of archimedean ℓ-groups with designated weak order unit. The full subcategory of W of objects for which the unit is strong unit is denoted by W* ; such ℓ-groups are called bounded. Thus arises a core ectionW B/→ W* . For G ∈ W, YG is the Yosida space, and G ≤pG is the much-studied projectable hull. Recently, in [1], for G ∈W, Y pG is identied as the Stone space of a certain boolean algebra A (G) of subsets of the minimal prime spectrum Min(G), and skepticism is expressed about extending this to W. Here, we show that indeed such an extension is possible, using a result from [5] and the following simple facts: in very concrete ways Min(G) and Min(BG) are homeomorphic spaces, and A (G) and A (BG) are isomorphic boolean algebras; p and B commute.

Mathematics Subject Classication (2010): Primary: 06F20, 46A40; Secondary: 54D35.

Key words: Archimedean ℓ-group, vector lattice, Yosida representation, minimal prime
spectrum, principal polar, projectable, principal projection property.