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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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References:
 [1] G. Birkhoff, J. von Neumann: The logic of quantum mechanics. Annals of Mathematics, 37 (1936), 823-843. · Zbl 0015.14603 [2] J. Brabec: Compatibility in orthomodular posets. Čas. pěst. mat., 104 (1979), 149-153. · Zbl 0416.06004 [3] J. Brabec: On some mappings generating vector L-measures. Čas. pěst. mat. 106, (1981), 347-353. · Zbl 0482.28017 [4] J. Brabec, P. Pták: On compatibility in quantum logics. Foundations of Physics 72, No. 2, (1982), 207-212. [5] R. Greechie: Orthomodular lattices admitting no states. Journ. Comb. Theory 10, (1971), 119-132. · Zbl 0219.06007 [6] S. Gudder: Spectral methods for a generalized probability theory. Trans. Amer. Math. Soc. 119, (1966), 428-442. · Zbl 0161.46105 [7] S. Gudder: Stochastic Methods in Quantum Mechanics. Elsevier North Holland, Inc., 1979. · Zbl 0439.46047 [8] K. Kuratowski: Topology I. Academic Press, 1966. · Zbl 0158.40901 [9] T. Neubrunn S. Pulmannovd: Compatibility in quantum logics. to appear in Acta Fac. Rev. Natur. Univ. Comen. Math. 1982. [10] T. Neubrunn: A note on the quantum probability spaces. Proc. Amer. Math. Soc. 25, no. 3, 1970, 672-675. · Zbl 0208.43402 [11] P. Pták: An observation on observables. to appear in Acta Polytechnica 1984. [12] P. Pták: Realcompactness and the notion of observable. J. London Math. Soc. 2, 23, (1981), 534-536. · Zbl 0447.54035 [13] P. Pták, V. Rogalewicz: Regularly full logics and the uniqueness problem for observables. Ann. Inst. Henri Poincaré, Vol. XXXVIII, no. 1, (1983), 69-74. · Zbl 0519.03051 [14] R. Sikorski: On the inducing of homomorphisms by mappings. Fund. Math. 36, (1947), 7-22. · Zbl 0040.17001 [15] M. Šulista: Observables and scales. (Czech), Habilitation, Technical University of Prague, 1974. [16] S. Ulam: Zur Masstheorie in der allgemeinen Mengenlehre. Fund. Math. 16, (1930), 140-150. · JFM 56.0920.04 [17] V. Varadarajan: Geometry of Quantum Theory. Vol. 1, Princeton, New Jersey: Van Nostrand Reinhold, 1968. · Zbl 0155.56802
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