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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.

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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