The Boolean transfer principle for injective Banach lattices

The aim of this paper is to apply the approach of Boolean-valued analysis to the theory of injective Banach lattices and to establish the Boolean-valued transfer principle from $AL$-spaces to injective Banach lattices. We prove that each injective Banach lattice embeds into an appropriate Boolean-valued model, becoming an $AL$-space. Hence, each theorem about an $AL$-space within Zermelo–Fraenkel set theory has an analog in the original injective Banach lattice interpreted as a Boolean-valued $AL$-space. Translation of theorems from $AL$-spaces to injective Banach lattices is carried out by the appropriate general operations of Boolean-valued analysis.