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

TL;DR

This paper explores lattice constructions in operator algebras to relate injective envelopes of C*-algebras with their von Neumann algebra counterparts, and investigates projection lattices and nonlinear maps in this context.

Contribution

It introduces lattice-based methods to connect injective envelopes with von Neumann algebras and analyzes projection lattices and nonlinear projection maps in operator algebras.

Findings

Established relations between injective envelopes and von Neumann algebras.

Analyzed lattices of projections in injective C*-algebras and von Neumann algebras.

Characterized nonlinear maps sending projections to projections.

Abstract

The article associates two fundamental lattice constructions with each regular unital real ordered Banach space (function system). These are used to establish certain results in the theory of operator algebras, specifically relating the injective envelope of a separable C*-algebra with its enveloping von Neumann algebra in a given faithful separable representation. The last section investigates on lattices of projections arising in injective C*-algebras and von Neumann algebras and certain nonlinear maps sending projections to projections which are essentially determined by their values on positive projections.