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A spectral theorem for \(\sigma \)-MV-algebras. (English) Zbl 1249.03119

Summary: MV-algebras were introduced by C. C. Chang [Trans. Am. Math. Soc. 88, 467–490 (1958; Zbl 0084.00704)] as algebraic bases for multi-valued logic. MV stands for “multi-valued” and MV-algebras have already occupied an important place in the realm of nonstandard (mathematical) logic applied in several fields including cybernetics. In the present paper, using the Loomis-Sikorski theorem for \(\sigma \)-MV-algebras, we prove that, with every element a in a \(\sigma \)-MV-algebra \(M\), a spectral measure (i.e. an observable) \(\Lambda \: \mathcal {B}([0,1])\to \mathcal {B}(M)\) can be associated, where \(\mathcal {B}(M)\) denotes the Boolean \(\sigma \)-algebra of idempotent elements in \(M\). This result is similar to the spectral theorem for self-adjoint operators on a Hilbert space. We also prove that MV-algebra operations are reflected by the functional calculus of observables.

MSC:

03G12 Quantum logic
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)

Citations:

Zbl 0084.00704
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References:

[1] Belluce L. P.: Semisimple algebras of infinite valued logic and Bold fuzzy set theory. Canad. J. Math. 38 (1986), 1356-1379 · Zbl 0625.03009 · doi:10.4153/CJM-1986-069-0
[2] Busch P., Lahti P. J., Mittelstaedt P.: The Quantum Theory of Measurement. Springer-Verlag, Berlin 1991 · Zbl 0868.46051 · doi:10.1007/978-3-540-37205-9
[3] Butnariu D., Klement E.: Triangular-norm-based measures and their Markov kernel representation. J. Math. Anal. Appl. 162 (1991), 111-143 · Zbl 0751.60003 · doi:10.1016/0022-247X(91)90181-X
[4] Barbieri G., Weber H.: Measures on clans and on MV-algebras. Handbook of Measure Theory, Vol. II (E. Pap, Elsevier, Amsterdam 2002, Chapt. 22, pp. 911-945 · Zbl 1019.28009
[5] Cattaneo G., Giuntini, R., Pulmannová S.: Pre-BZ and degenerate BZ posets: Applications to fuzzy sets and unsharp quantum theories. Found. Phys. 30 (2000), 1765-1799 · doi:10.1023/A:1026462620062
[6] Chang C. C.: Algebraic analysis of many-valued logics. Trans. Amer. Math. Soc. 88 (1958), 467-490 · Zbl 0084.00704 · doi:10.2307/1993227
[7] Chang C. C.: A new proof of the completeness of the Lukasiewicz axioms. Trans. Amer. Math. Soc. 93 (1959), 74-80 · Zbl 0093.01104 · doi:10.2307/1993423
[8] Chovanec F., Kôpka F.: D-lattices. Internat. J. Theor. Phys. 34 (1995), 1297-1302 · Zbl 0840.03046 · doi:10.1007/BF00676241
[9] Cignoli R., D’Ottaviano I. M. L., Mundici D.: Algebraic Foundation of Many-Valued Reasoning. Kluwer Academic Publishers, Dordrecht 2000 · Zbl 0937.06009
[10] Nola A. Di, Dvurečenskij A., Hyčko, M., Manara C.: Entropy on effect algebras with the Riesz decomposition property I, II. Kybernetika 41 (2005), 143-160, 161-176 · Zbl 1249.03115
[11] Chiara M. Dalla, Giuntini, R., Greechie R.: Reasoning in Quantum Theory. Kluwer Academic Publishers, Dordrecht 2004 · Zbl 1059.81003
[12] 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 · doi:10.1017/S1446788700001993
[13] Dvurečenskij A., Pulmannová S.: New Trends in Quantum Structures. Kluwer Academic Publishers, Dordrecht and Ister Science, Bratislava 2000 · Zbl 0987.81005
[14] Foulis D. J., Bennett M. K.: Effect algebras and unsharp quantum logic. Found. Phys. 24 (1994), 1325-1346 · Zbl 1213.06004 · doi:10.1007/BF02283036
[15] Halmos P. R.: Measure Theory. Van Nostrand, Princeton, New Jersey 1962 · Zbl 0283.28001
[16] Kôpka F., Chovanec F.: D-posets. Math. Slovaca 44 (1994), 21-34 · Zbl 0789.03048
[17] Mundici D.: Interpretation of AF C*-algebras in Lukasiewicz sentential calculus. J. Funct. Anal. 65 (1986), 15-63 · Zbl 0597.46059 · doi:10.1016/0022-1236(86)90015-7
[18] Mundici D.: Tensor products and the Loomis-Sikorski theorem for MV-algebras. Adv. Appl. Math. 22 (1999), 227-248 · Zbl 0926.06004 · doi:10.1006/aama.1998.0631
[19] Pták P., Pulmannová S.: Orthomodular Structures as Quantum Logics. Kluwer Academic Publishers, Dordrecht and VEDA, Bratislava 1991 · Zbl 0743.03039
[20] Pulmannová S.: Spectral resolutions in Dedekind \(\sigma \)-complete \(\ell \)-groups. J. Math. Anal. Appl. · Zbl 1072.06014 · doi:10.1016/j.jmaa.2005.01.044
[21] Riečan B., Mundici D.: Probability on MV-algebras. Handbook of Measure Theory, Vol. II (E. Pap, Elsevier, Amsterdam 2002, Chapt. 21, pp. 869-909 · Zbl 1017.28002
[22] Riečan B., Neubrunn T.: Integral, Measure and Ordering. Kluwer Academic Publishers, Dordrecht and Ister Science, Bratislava 1997 · Zbl 0916.28001
[23] Varadarajan V. S.: Geometry of Quantum Theory. Springer-Verlag, New York 1985 · Zbl 0581.46061
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