×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] ALFSEN E.-SCHULTZ F.: On the geometry of noncommutative spectral theory. Bull. Amer. Math. Soc. 81 (1975), 893-895. · Zbl 0337.46014
[2] BENNETT M. K.-FOULIS D. J.: Interval and scale effect algebras. Adv. in Appl. Math. 19 (1997), 200-215. · Zbl 0883.03048
[3] BLYTH T. S.-JANOWITZ M. F.: Residuation Theory. Pergamon, New York, 1972. · Zbl 0301.06001
[4] FOULIS D. J.: MV and Heyting effect algebras. Found. Phys. 30 (2000), 1687-1706.
[5] FOULIS D. J.: Compressions on partially ordered abelian groups. · Zbl 1063.47003
[6] FOULIS D. J.: Removing the torsion from a unital group. Rep. Math. Phys. · Zbl 1054.81005
[7] FOULIS D. J.-GREECHIE R. J.-BENNETT M. K.: The transition to unigroups. Internat. J. Theoret. Phys. 37 (1998), 45-64. · Zbl 0904.06013
[8] GREECHIE R. J.-FOULIS D. J.-PULMANNOVÁ S.: The center of an effect algebra. Order 12 (1995), 91-106. · Zbl 0846.03031
[9] GOODEARL K. R.: Partially Ordered Abelian Groups with Interpolation. Math. Surveys Monographs 20, Amer. Math. Soc, Providence, RI, 1986. · Zbl 0589.06008
[10] GUDDER S. P.: Examples, problems, and results in effect algebras. Internat. J. Theoret. Phys. 35 (1996), 2365-2376. · Zbl 0868.03028
[11] GUDDER S. P.: Sharply dominating effect algebras. Tatra Mt. Math. Publ. 15 (1998), 23-30. · Zbl 0939.03073
[12] GUDDER S. P.-PULMANNOVÁ S.-BUGAJSKI S.-BELTRAMETTI E. G.: Convex and linear effect algebras. Rep. Math. Phys. 44 (1999), 359-379. · Zbl 0956.46002
[13] HANDELMAN, D-HIGGS D.-LAWRENCE J.: Directed abelian groups, countably continuous rings, and Rickart \(C^\ast\) -algebras. J. London Math. Soc 21 (1980), 193-202. · Zbl 0449.06013
[14] JENČA G.: Blocks of homogeneous effect algebras. Bull. Austral. Math. Soc. 64 (2001), 81-98. · Zbl 0985.03063
[15] KADISON R. V.: Order properties of bounded self-adjoint operators. Proc. Amer. Math. Soc 2 (1951), 505-510. · Zbl 0043.11501
[16] PTÁK P.-PULMANNOVÁ S.: Orthomodular Structures as Quantum Logics. Kluwer, Dordrecht-Boston-London, 1991. · Zbl 0743.03039
[17] PULMANNOVÁ S.: Effect algebras with the Riesz decomposition property and \(AF\) \(C^\ast\)-algebras. Found. Phys. 29 (1999), 1389-1401.
[18] RIESZ F.-SZ.-NAGY B.: Functional Analysis. Frederick Ungar Publishing Co., New York, 1955.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.