Abstract
It is shown that any order isomorphism between the structures of unital associative JB subalgebras of JB algebras is given naturally by a partially linear Jordan isomorphism. The same holds for nonunital subalgebras and order isomorphisms preserving the unital subalgebra. Finally, we recover usual action of time evolution group on a von Neumann factor from group of automorphisms of the structure of Abelian subalgebras.
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References
Bunce, L.J., Wright, J.D.M.: The Mackey Gleason problem. Bull. Am. Math. Soc. 26(2), 288–293 (1992)
Connes, A.: A factor not isomorphic to itself. Ann. Math. 101(3), 536–554 (1962)
Döring, A., Isham, C.J.: A topos foundation for theories of physics: IV. Categories of systems. J. Math. Phys. 49, 953518 (2008)
Döring, A., Harding, J.: Abelian subalgebras and the Jordan structure of von Neuman algebras. arXiv:1009.4945v1
Döring, A.: Generalised Gelfand Spectra of Nonabelian Unital C ∗-Algebras I: Categorical Aspects, Automorphisms and Jordan Structure. arXiv:1212.2613
Döring, A.: Generalised Gelfand Spectra of Nonabelian Unital C ∗-Algebras II: Flows and Time Evolution of Quantum Systems. arXiv:1212.4882
Emch, G.G.: Algebraic Methods in Statistical Mechanics and Quantum Field Theory. Dover, New York (2009)
Hamhalter, J.: Quantum Measure Theory. Kluwer Academic, Dordrecht (2003)
Hamhalter, J.: Isomorphisms of ordered structures of Abelian C ∗-subalgebras of C ∗-algebras. J. Math. Anal. Appl. 383, 391–399 (2011)
Hamhalter, J., Turilova, E.: Structure of associative subalgebras of Jordan operator algebras. Q. J. Math. 00, 1–12 (2012). doi:10.1093/qmath/has015
Heunen, C., Landsman, N.P., Spitters, B.: Bohrification of operator algebras and quantum logic. Halvorson, H. (ed.): Deep Beauty Understanding the Quantum World Through Mathematical Innovation, Princeton University, pp. 217–313
Heunen, C.: Characterizations of categories of commutative C ∗-subalgebras (2012). arXiv:1106.5942v2
Heunen, C., Reyes, M.L.: Active lattices determine AW∗-algebras (2012). arXiv:1212.5778v1
Hanche-Olsen, H., Stormer, E.: Jordan Operator Algebras. Pitman, London (1984)
Strocchi, F.: An Introduction to the Mathematical Structure of Quantum Mechanics. Advanced Series in Mathematical Physics, vol. 28. World Scientific, Singapore (2008)
Takesaki, M.: Theory of Operator Algebras I. Springer, New York (1979)
Takesaki, M.: Theory of Operator Algebras II. Springer, New York (2003)
Weaver, N.: Mathematical Quantization. Studies in Advanced Mathematics. Chapman & Hall, London (2001)
Acknowledgements
This work was supported by the ‘Grant Agency of the Czech Republic’ grant number P201/12/0290, “Topological and geometrical properties of Banach spaces and operator algebras”.
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Hamhalter, J., Turilova, E. Automorphisms of Order Structures of Abelian Parts of Operator Algebras and Their Role in Quantum Theory. Int J Theor Phys 53, 3333–3345 (2014). https://doi.org/10.1007/s10773-013-1691-3
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DOI: https://doi.org/10.1007/s10773-013-1691-3