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Spaces of observables. (English) Zbl 0572.28007
The main question considered in the paper is when (and how) one can introduce a linear and topological structure into sets of observables. (An observable is defined as a \(\sigma\)-homomorphism from the \(\tau\)- algebra of Borel sets of a Banach space into a \(\sigma\)-orthomodular poset). The first part of the paper brings the result that certain sets of observables can be converted to Banach spaces. This generalizes the result by S. Gudder [Trans. Am. Math. Soc. 119, 428-442 (1965; Zbl 0161.461)]. Then one introduces compact observables as an extension of real bounded observables. Finally, for Banach spaces possibly nonseparable, one indicates obstacles which lie on the way to ”linearizing” observables.

MSC:
28B15 Set functions, measures and integrals with values in ordered spaces
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
03G12 Quantum logic
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