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.


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


Zbl 0084.00704
Full Text: EuDML Link


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