Abstract
It is shown that the unit interval of a von Neumann algebra is a Sum Brouwer–Zadeh algebra when equipped with another unary operation sending each element to the complement of its range projection. The main result of this Letter says that a von Neumann algebra is finite if and only if the corresponding Brouwer–Zadeh structure is de Morgan or, equivalently, if the range projection map preserves infima in the unit interval. This provides a new characterization of finiteness in the Murray–von Neumann structure theory of von Neumann algebras in terms of Brouwer–Zadeh structures.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Busch, P., Grabowski, M. and Lahti, P. J.: Operational Quantum Physics, Lecture Notes in Phys. Springer-Verlag, Berlin, 1995.
Cattaneo, G.: A unified framework for the algebra of unsharp quantum mechanics, Internat. J. Theoret. Phys. 36 (1997), 3085–3117.
Cattaneo, G. and Nisticò, G.: Brouwer-Zadeh posets and three valued ?ukasiewicz posets, Fuzzy Sets and Systems 33 (1989), 165–190.
Cattaneo, G. and Giuntini, R.: Some results on BZ structures from Hilbertian unsharp quantum physics, Found. Phys. 25 (1995), 1147–1183.
Dalla Chiara, M. L. and Giuntini, R.: Unsharp quantum logics, Found. Phys. 24 (1994), 1161–1177.
Dvurecenskij, A.: Measures and ?-decomposable measures on effect algebras of a Hilbért space, Atti Sem. Mat. Fis. Univ. Modena 45 (1997), 259–288.
Foulis, D. J. and Bennett, M. K.: Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994), 1325–1346.
Foulis, D. J. and Randall, C.: Empirical logic and tensor product, In: H. Neumann (ed.), Interpretation and Foundations of Quantum Mechanics, Wissenschaftsverlag, Bibliographisches Institut, Mannheim, 1981, pp. 9–20.
Gudder, S.: Semi-orthoposets, Internat. J. Theoret. Phys. 35 (1996), 1141–1173.
Gudder, S.: Sharply dominating effect algebras, Tatra Mt. Math. Publ. 15 (1998), 23–30; Special Issue: Quantum Structures II, Dedicated to Gudrun Kalmbach.
Kadison, R. V. and Ringrose, J. R.: Fundamentals of the Theory of Operator Algebras II, Academic Press, New York, 1983.
Mackey, G. W.: The Mathematical Foundations of Quantum Mechanics, Benjamin, New York, 1963.
Murphy, G. J.: C*-Algebras and Operator Theory, Academic Press, New York, 1990.
Pedersen G. K.: C*-Algebras and their Automorphism Groups, Academic Press, New York, 1989.
Segal, I. E.: Postulates for general quantum mechanics, Ann. Math. 48 (1947), 830–948.
Stratila, S. and Szidó, L.: Lectures on von Neumann Algebras, Abacus Press, England, 1979.
Takesaki, M.: Theory of Operator Algebras I, Springer-Verlag, New York, 1971.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cattaneo, G., Hamhalter, J. De Morgan Property for Effect Algebras of von Neumann Algebras. Letters in Mathematical Physics 59, 243–252 (2002). https://doi.org/10.1023/A:1015584530597
Issue Date:
DOI: https://doi.org/10.1023/A:1015584530597