zbMATH — the first resource for mathematics

Spectral resolutions for $$\sigma$$-complete lattice effect algebras. (English) Zbl 1141.81007
By a theorem of Z. Riečanová [Int. J. Theor. Phys. 39, No. 2, 231–237 (2000; Zbl 0968.81003)], every lattice-ordered effect algebra is a union of sub-effect algebras called blocks, and each block is an MV-algebra. The author has obtained a spectral theorem for $$\sigma$$-MV-algebras [Kybernetika 41, No. 3, 361–374 (2005); J. Math. Anal. Appl. 309, No. 1, 322–335 (2005; Zbl 1072.06014)]. In the present paper, the author is able to combine these two results to obtain spectral resolutions for elements in a $$\sigma$$-complete lattice-ordered effect algebra $$E$$. This is accomplished by showing that the blocks in $$E$$ are $$\sigma$$-complete MV-algebras, and that if an element of $$E$$ belongs to two different blocks, then its spectral resolutions, as calculated in the two blocks, must coincide. Section 1 of the paper is an excellent overview of known results, many due to the author herself, pertaining to $$\sigma$$-MV-algebras. States and observables have roles to play in the paper, hence the theory developed here has potential applications to physics. For instance, the set Sh$$(E)$$ of all sharp elements of $$E$$ is a $$\sigma$$-complete orthomodular lattice (analogous to the lattice of projections on a Hilbert space), and it is shown that (by analogy with the celebrated Gleason theorem), every $$\sigma$$-state on Sh$$(E)$$ extends uniquely to a $$\sigma$$-state on $$E$$ in such a way that pure states extend to pure states.

MSC:
 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 03G12 Quantum logic
Citations:
Zbl 0968.81003; Zbl 1072.06014
Full Text:
References:
 [1] BARBIERI G.-WEBER H.: Measures on clans and on MV-algebras. Handbook of Measure Theoru, vol II (E. Pap, Elsevier, Amsterdam, 2002, pp. 911-945 · Zbl 1019.28009 [2] BUTNARIU D.-KLEMENT E.: Triangular-norm-based measures and their Markov kernel representation. J. Math. Anal. Appl. 162 (1991), 111-143. · Zbl 0751.60003 [3] CHANG C. C.: Algebraic analysis of many-valued logics. Trans. Amer. Math. Soc. 88 (1958), 467-490. · Zbl 0084.00704 [4] CHANG C. C.: A new proof of the completeness of the Lukasiewicz axioms. Trans. Amer. Math. Soc. 93 (1959), 74-80. · Zbl 0093.01104 [5] CIGNOLI R.-D’OTTAVIANO I. M. L.-MUNDICI D.: Algebraic Foundation of Many-Valued Reasoning. Kluwer Acad. PubL, Dordrecht, 2000. · Zbl 0937.06009 [6] CHOVANEC F.-KOPKA F.: D-lattices. Internat. J. Theoret. Phys. 34 (1995), 1297-1302. · Zbl 0840.03046 [7] DVUREČENSKIJ A.: Loomis-Sikorski theorem for $$\sigma$$-complete $$MV$$-algebras and \ell$$-groups. J. Austral. Math. Soc. Ser. A 68 (2000), 261-277.$$ · Zbl 0958.06006 [8] DVUREČENSKIJ A.: MV-observables and MV-algebras. J. Math. Anal. Appl. 259 (2001), 413-428. · Zbl 0992.03081 [9] DVUREČENSKIJ A.-PULMANNOVÁ S.: New Trends in Quantum Structures. Kluwer Acad. Publ./Ister Science, Dordrecht/Bratislava, 2000. · Zbl 0987.81005 [10] DVUREČENSKIJ A.-PULMANNOVÁ S.: Conditional probability on a-MV algebras. Fuzzy Sets and Systems 155 (2005), 102-118. · Zbl 1080.06012 [11] FOULIS D. J.-BENNETT M. K.: Effect algebras and unsharp quantum logic. Found. Phys. 24 (1994), 1325-1346. · Zbl 1213.06004 [12] FOULIS D. J.: Compressible groups. Math. Slovaca 53 (2003), 433-455. · Zbl 1114.06012 [13] FOULIS D. J.: Compressions on partially ordered abelian groups. Proc. Amer. Math.Soc. 132 (2004), 3581-3587. · Zbl 1063.47003 [14] FOULIS D. J.: Spectral resolution in a Rickart comgroup. Rep. Math. Phys. 54 (2004), 319-340. · Zbl 1161.81310 [15] FOULIS D. J.: Compressible groups with general comparability. Math. Slovaca. 55 (2005), 409-429. · Zbl 1114.06012 [16] FOULIS D. J.: MV and Heyting effect algebras. Found. Phys. 30 (2000), 1687-1706. [17] GIUNTINI R.-GREULING H.: Toward a formal language for unsharp properties. Found. Phys. 19 (1989), 931-945. [18] GOODEARL K. R.: Partially Ordered Abelian Groups with Interpolation. Math. Surveys Monogr. 20, Amer. Math. Soc, Providence, RI, 1986. · Zbl 0589.06008 [19] GUDDER S. P.: S-dominating effect algebras. Internat. J. Theoret. Phys. 37 (1998), 915-923. · Zbl 0932.03072 [20] GUDDER S. P.: Compressible effect algebras. Rep. Math. Phys. 54 (2004), 93-114. · Zbl 1075.81011 [21] JENČA G.: Sharp and meager elements in orthocomplete homogeneous effect algebras. Preprint, 2004 (Available from · Zbl 1193.03084 [22] JENČA G.-PULMANNOVÁ S.: Orthocomplete effect algebras. Proc Amer. Math.Soc 131 (2003), 2663-2671. · Zbl 1019.03046 [23] JENČA G.-PULMANNOVÁ S.: Ideals and quotients in lattice ordered effect algebras. Soft Comput. 5 (2001), 376-380. · Zbl 1004.06009 [24] JENČA G.-RIEČANOVÁ Z.: On sharp elements in lattice ordered effect algebras. Busefal 80 (1999), 24-29. [25] KOPKA F.-CHOVANEC F.: D-posets. Math. Slovaca 44 (1994), 21-34. · Zbl 0789.03048 [26] MUNDICI D.: Interpretations of $$AF$$ $$C^\ast$$-algebras in Lukasziewicz sentential calculus. J. Funct. Anal. 65 (1986), 15-63. · Zbl 0597.46059 [27] MUNDICI D.: Tensor products and the Loomis-Sikorski theorem for MV-algebras. Adv. in Appl. Math. 22 (1999), 227-248. · Zbl 0926.06004 [28] PULMANNOVÁ S.: A spectral theorem for a-MV algebras. Kybernetika 41 (2005), 361-374. · Zbl 1249.03119 [29] PULMANNOVÁ S.: Spectral resolutions in Dedekind $$\sigma$$-complete $$\ell$$-groups. J. Math. Anal. Appl. 309 (2005), 322-335. · Zbl 1072.06014 [30] PULMANNOVÁ S.: Commutator-finite D-lattices. Order 21 (2004), 91-105. · Zbl 1081.06013 [31] PTÁK P.-PULMANNOVÁ S.: Orthomodular Structures as Quantum Logics. Kluwer Acad. Publ./VEDA, Dordrecht/Bratislava, 1991. · Zbl 0743.03039 [32] RIEČAN B.-NEUBRUNN T.: Integral, Measure and Ordering. Kluwer Acad. Publ./ Ister Science, Dordrecht/Bratislava, 1997. · Zbl 0916.28001 [33] RIEČANOVÁ Z.: Generalization of blocks for D-lattices and lattice ordered effect algebras. Internat. J. Theoret. Phys. 39 (2000), 231-237. · Zbl 0968.81003 [34] RIEČANOVÁ Z.: Smearing of states defined on sharp elements onto effect algebras. Internat. J. Theoret. Phys. 41 (2002), 1511-1524. · Zbl 1016.81005 [35] VARADARAJAN V. S.: Geometry of Quantum Theory. Springer-Verlag, New York, 1985. · Zbl 0581.46061
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.