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Two-dimensional observables and spectral resolutions. (English) Zbl 1441.81004
Summary: A two-dimensional observable is a special kind of a $$\sigma$$-homomorphism defined on the Borel $$\sigma$$-algebra of the real plane with values in a $$\sigma$$-complete MV-algebra or in a monotone $$\sigma$$-complete effect algebra. A two-dimensional spectral resolution is a mapping defined on the real plane with values in a $$\sigma$$-complete MV-algebra or in a monotone $$\sigma$$-complete effect algebra which has properties similar to a two-dimensional distribution function in probability theory. We show that there is a one-to-one correspondence between two-dimensional observables and two-dimensional spectral resolutions defined on a $$\sigma$$-complete MV-algebras as well as on the monotone $$\sigma$$-complete effect algebras with the Riesz decomposition property. The result is applied to the existence of a joint two-dimensional observable of two one-dimensional observables.
##### MSC:
 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 28A05 Classes of sets (Borel fields, $$\sigma$$-rings, etc.), measurable sets, Suslin sets, analytic sets 06D35 MV-algebras
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