# zbMATH — the first resource for mathematics

Representations of MV-algebras by Hilbert-space effects. (English) Zbl 1270.06006
Summary: It is shown that for every Archimedean MV-effect algebra $$M$$ (equivalently, every Archimedean MV-algebra) there is an injective MV-algebra morphism into the MV-algebra of all multiplication operators between the zero and identity operator on $$\ell_{2}(\mathcal{S}_{0})$$, where $$\mathcal{S}_{0}$$ is an ordering set of extremal states (state morphisms) on $$M$$.

##### MSC:
 06D35 MV-algebras 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
Full Text:
##### References:
 [1] Bennett, M.K.; Foulis, D.J., Phi-symmetric effect algebras, Found. Phys., 25, 1699-1722, (1995) [2] Cattaneo, G.; Giuntini, R.; Pulmannová, S., Pre-BZ and degenerate BZ-posets: applications to fuzzy sets and unsharp quantum theories, Found. Phys., 30, 1765-1799, (2000) [3] Chang, C.C., Algebraic analysis of many-valued logics, Trans. Am. Math. Soc., 88, 467-490, (1958) · Zbl 0084.00704 [4] Chovanec, F.; Kôpka, F., D-lattices, Int. J. Theor. Phys., 34, 1297-1302, (1995) · Zbl 0840.03046 [5] Cignoli, R., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Kluwer, Dordrecht (2000) · Zbl 0937.06009 [6] Dvurečeskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer, Dordrecht (2000) [7] Foulis, D.; Bennett, M.K., Effect algebras and unsharp quantum logics, Found. Phys., 24, 1325-1346, (1994) · Zbl 1213.06004 [8] Giuntini, R.; Greuling, H., Toward a formal language for unsharp properties, Found. Phys., 19, 931-945, (1989) [9] Gudder, S.P., Effect algebras are not adequate models for quantum mechanics, Found. Phys., 40, 1566-1577, (2010) · Zbl 1218.81010 [10] Jenča, G.; Pulmannová, S., Orthocomplete effect algebras, Proc. Am. Math. Soc., 131, 2663-2671, (2003) · Zbl 1019.03046 [11] Kadison, R.V., Order properties of bounded self-adjoint operators, Proc. AMS, 2, 506-510, (1951) · Zbl 0043.11501 [12] Kôpka, F.; Chovanec, F., D-posets, Math. Slovaca, 44, 21-34, (1994) · Zbl 0789.03048 [13] Mundici, D., Interpretation of AF C*-algebras in łukasiewicz sentential calculus, J. Funct. Anal., 65, 15-63, (1986) · Zbl 0597.46059 [14] Pulmannová, S., Compatibility and decompositions of effects, J. Math. Phys., 43, 2817-2830, (2002) · Zbl 1059.81016 [15] Pulmannová, S., On fuzzy hidden variables, Fuzzy Sets Syst., 155, 119-137, (2005) · Zbl 1079.81008 [16] Riečanová, Z.; Zajac, M., Hilbert-space effect-representations of effect algebras, Rep. Math. Phys., 40, 1566-1575, (2010) [17] Varadarajan, V.S.: Geometry of Quantum Theory. Springer, 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.