A noncommutative version of a Theorem of Marczewski for submeasures

Paolo de Lucia; Pedro Morales

ArticleOriginal scientific text

Title

A noncommutative version of a Theorem of Marczewski for submeasures

Authors ¹, ²

Affiliations

Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli, Via Mezzocannone, 8, I-80134 Napoli, Italy
Département de Mathématiques et d'Informatique, Université de Sherbrooke, Sherbrooke, Québec, Canada J1K 2R1

Abstract

It is shown that every monocompact submeasure on an orthomodular poset is order continuous. From this generalization of the classical Marczewski Theorem, several results of commutative Measure Theory are derived and unified.

A. D. Alexandroff [A. D. Aleksandrov], Additive set-functions in abstract spaces I, Mat. Sb. 8 (50) (1940), 307-348.
A. D. Alexandroff [A. D. Aleksandrov], Additive set functions in abstract spaces II, ibid. 9 (51) (1941), 563-628.
O. R. Béaver and T. A. Cook, States on quantum logics and their connection with a theorem of Alexandroff, Proc. Amer. Math. Soc. 67 (1977), 133-134.
L. Beran, Orthomodular Lattices-Algebraic Approach, Academia, Praha 1984.
K. P. S. Bhaskara Rao and M. Bhaskara Rao, Theory of Charges, Academic Press, London 1983.
N. Bourbaki, Topologie générale, 3rd ed., Actualités Sci. Indust. 1143, Chaps. 3 and 4, Hermann, Paris 1960.
L. Drewnowski, Topological rings of sets, continuous set functions, integration. II, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), 277-286.
N. Dunford and J. Schwartz, Linear Operators I, Interscience, New York 1958.
R. Engelking, General Topology, Polish Scientific Publishers, Warszawa 1977.
W. Gähler, Grundstrukturen der Analysis, Vol. I, Akademie-Verlag, Berlin 1977.
L. Gillman and M. Jerison, Rings of Continuous Functions, Springer, New York 1976.
A. Gleason, Measures on the closed subspaces of a Hilbert space, J. Math. Mech. 6 (1957), 885-893.
I. Glicksberg, The representation of functionals by integrals, Duke Math. J. 19 (1952), 253-261.
J. E. Huneycutt, Jr., Extensions of abstract valued set functions, Trans. Amer. Math. Soc. 141 (1969), 505-513.
G. Kalmbach, Orthomodular Lattices, Academic Press, London 1983.
G. Kalmbach, Measures and Hilbert Lattices, World Scientific, Singapore 1986.
I. Kluvánek, Integration Structures, Proc. Centre Math. Anal. Austral. Nat. Univ. 18, 1988.
K. Kuratowski, Topology I, Academic Press, London 1966.
E. Marczewski, On compact measures, Fund. Math. 40 (1953), 113-124.
H. Millington, Products of group-valued measures, Studia Math. 54 (1975), 7-27.
P. Morales, Regularity and extension of semigroup-valued Baire measures, in: Proc. Conf. Measure Theory, Oberwolfach 1979, Lecture Notes in Math. 794, Springer, New York 1980, 317-323.
J. von Neumann, Functional Operators I, Princeton Univ. Press, Princeton, N.J., 1950.
J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton Univ. Press, Princeton, N.J., 1955.
G. T. Rüttimann, Non-commutative measure theory, preprint, University of Berne, 1979.
R. Schatten, Norm Ideals of Completely Continuous Operators, Springer, Berlin 1960.
J. Šipoš, Subalgebras and sublogics of σ-logics, Math. Slovaca 28 (1) (1978), 3-9.
R. M. Stephenson, Pseudo-compact spaces, Trans. Amer. Math. Soc. 134 (1968), 437-448.
A. Sudbery, Quantum Mechanics and the Particles of Nature, Cambridge Univ. Press, Cambridge 1986.
K. Sundaresan and P. W. Day, Regularity of group valued Baire and Borel measures, Proc. Amer. Math. Soc. 36 (1972), 609-612.
F. Topsøe, Approximating pavings and construction of measures, Colloq. Math. 42 (1979), 377-385.
V. S. Varadarajan, Geometry of Quantum Theory, 2nd ed., Springer, Berlin 1985.

Title

A noncommutative version of a Theorem of Marczewski for submeasures

Affiliations

Abstract

Bibliography