Kantorovich’s Principle in Action: Aw ∗-modules and Injective Banach Lattices

The aim of this note is to demonstrate that Kaplansky–Hilbert lattices and injective Banach lattices may be produced from each other by means of the well known convexification procedure. This is done via the Boolean valued analysis approach. The subject gives a good opportunity to discuss also the relationship between the Kantorovich’s heuristic principle and the Boolean value transfer principle. Everywhere below B is a complete Boolean algebra and V(B) the corresponding Boolean valued model of set theory, see [3, 22]. Let Λ be a real Dedekind complete AM -space with unit endowed with a unique f -algebra multiplication. Then Λ̄ := Λ ⊕ iΛ is a commutative C∗-algebra often called a Stone algebra. We write Λ = Λ(B) whenever B is a Boolean algebra of band projections in Λ. The unexplained terms of use below can be found in [19] and [27].