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Compressible groups. (English) Zbl 1068.06018
Summary: We introduce and initiate a study of a new class of partially ordered abelian groups called compressible groups. The compressible groups generalize the order-unit space of self-adjoint operators on a Hilbert space, the directed additive group of self-adjoint elements of a unital $$C^*$$-algebra, lattice-ordered abelian groups with order unit, and interpolation groups with order unit. We identify elements, called projections, in a compressible group, show that the set $$P$$ of projections forms an orthomodular poset, and give sufficient conditions, satisfied in a Rickart $$C^*$$-algebra and in an interpolation group with order unit, for $$P$$ to form an orthomodular lattice.

##### MSC:
 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces 03G12 Quantum logic 46L05 General theory of $$C^*$$-algebras 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 06C15 Complemented lattices, orthocomplemented lattices and posets
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