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The article associates two fundamental lattice constructions with eachregular unital real ordered Banach space (function system). These are used toestablish certain results in the theory of operator algebras, specificallyrelating the injective envelope of a separable C*-algebra with its envelopingvon Neumann algebra in a given faithful separable representation. The lastsection investigates on lattices of projections arising in injectiveC*-algebras and von Neumann algebras and certain nonlinear maps sendingprojections to projections which are essentially determined by their values onpositive projections.

The Jordan lattice completion and a note on injective envelopes and von Neumann algebras