Abstract
A synaptic algebra A is a generalization of the self-adjoint part of a von Neumann algebra. We study a linear subspace V of A in regard to the question of when V is a vector lattice. Our main theorem states that if V contains the identity element of A and is closed under the formation of both the absolute value and the carrier of its elements, then V is a vector lattice if and only if the elements of V commute pairwise.
The second and third authors were supported by grant VEGA 2/0069/16.
References
[1] Alfsen, E. M.: Compact Convex Sets and Boundary Integrals, Springer-Verlag, New York, Heidelberg, Berlin, 1971.10.1007/978-3-642-65009-3Search in Google Scholar
[2] Beran, L.: Orthomodular Lattices. An Algebraic Approach. Mathematics and its Applications 18, D. Reidel Publishing Company, Dordrecht, 1985.10.1007/978-94-009-5215-7_5Search in Google Scholar
[3] Birkhoff, G.: Lattice Theory, Amer. Math. Soc. Colloq. Publ. XXV, Providence, RI, 1967.Search in Google Scholar
[4] Chang, C. C.: Algebraic analysis of many-valued logics, Trans. Amer. Math. Soc. 89 (1959), 74–80.10.1090/S0002-9947-1958-0094302-9Search in Google Scholar
[5] Foulis, D. J.: MV and Heyting effect algebras, Found. Phys. 30 (2000), 1687–1706.10.1023/A:1026454318245Search in Google Scholar
[6] Foulis, D. J.: Synaptic algebras, Math. Slovaca 60 (2010), 631–654.10.2478/s12175-010-0037-3Search in Google Scholar
[7] Foulis, D. J.—Bennett, M. K.: Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994), 1331–1352.10.1007/BF02283036Search in Google Scholar
[8] Foulis, D. J.—Pulmannová, S.: Monotone sigma-complete RC-groups, J. London Math. Soc. 73 (2006), 304–324.10.1112/S002461070602271XSearch in Google Scholar
[9] Foulis, D. J.—Pulmannová, S.: Projections in synaptic algebras, Order 27 (2010), 235–257.10.1007/s11083-010-9148-2Search in Google Scholar
[10] Foulis, D. J.—Pulmannová, S.: Type-decomposition of a synaptic algebra, Found. Phys. 43 (2013), 948–968.10.1007/s10701-013-9727-3Search in Google Scholar
[11] Foulis, D. J.—Pulmannová, S.: Symmetries in synaptic algebras, Math. Slovaca 64 (2014), 751–776.10.2478/s12175-014-0238-2Search in Google Scholar
[12] Foulis, D. J.—Pulmannová, S.: Commutativity in a synaptic algebra, Math. Slovaca 66 (2016), 469–482.10.1515/ms-2015-0151Search in Google Scholar
[13] Foulis, D. J.—Jenčová, A.—Pulmannová, S.: Two projections in a synaptic algebra, Linear Algebra Appl. 478 (2015), 162–187.10.1016/j.laa.2015.03.021Search in Google Scholar
[14] Foulis, D. J.—Jenčová, A.—Pulmannová, S.: A projection and an effect in a synaptic algebra, Linear Algebra Appl. 485 (2015), 417–441.10.1016/j.laa.2015.07.039Search in Google Scholar
[15] Foulis, D. J.—Jenčová, A.—Pulmannová, S.: Every synaptic algebra has the monotone square root property, Positivity 21 (2017), 919–930.10.1007/s11117-016-0443-zSearch in Google Scholar
[16] Goodearl, K. R.: Partially Ordered Abelian Groups with Interpolation. Math. Surveys Monogr. 20, A.M.S., 1980.Search in Google Scholar
[17] Gudder, S. P.: Lattice properties of quantum effects, J. Math. Phys. 37 (1996), 2637–2642.10.1063/1.531533Search in Google Scholar
[18] Jenča, G.: A representation theorem for MV-algebras, Soft Comput. 11 (2007), 557–564.10.1007/s00500-006-0100-8Search in Google Scholar
[19] Kadison, R. V.: Order properties of bounded self-adjoint operators, Proc. Amer. Math. Soc. 34 (1951), 505–510.10.1090/S0002-9939-1951-0042064-2Search in Google Scholar
[20] Kadison, R. V.—Ringrose, J. R.: Fundamentals of the Theory of Operator Algebras I: Elementary Theory, Academic Press, New York, 1983.Search in Google Scholar
[21] Kalmbach, G.: Orthomodular Lattices, Academic Press, London, New York, 1983.Search in Google Scholar
[22] Lahti, P.—Mączynski, M.: Partial order of quantum effects, J. Math. Phys. 36 (1995), 1673–1680.10.1063/1.531079Search in Google Scholar
[23] Pulmannová, S.: A note on ideals in synaptic algebras, Math. Slovaca 62 (2012), 1091–1104.10.2478/s12175-012-0067-0Search in Google Scholar
[24] Riečanová, Z.: Generalization of blocks for D-lattices and lattice-ordered effect algebras, Internat. J. Theoret. Phys. 39 (2000), 231–237.10.1023/A:1003619806024Search in Google Scholar
[25] Schaefer, H.: Banach Lattices and Positive Operators, Springer-Verlag, Berlin-Heidelberg-New York, 1974.10.1007/978-3-642-65970-6Search in Google Scholar
[26] Topping, D. M.: Vector lattices of self-adjoint operators, Trans. Amer. Math. Soc. 115 (1965), 14–30.10.1090/S0002-9947-1965-0206736-3Search in Google Scholar
© 2017 Mathematical Institute Slovak Academy of Sciences
