Spectral order on unbounded operators and their symmetries

J. Hamhalter^a, E.A. Turilova^b

^aCzech Technical University in Prague, Prague, 160 00 Czech Republic

^bKazan Federal University, Kazan, 420008 Russia

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Abstract

The spectral order on positive unbounded operators affiliated with von Neumann algebras have been considered. The spectral order has physical meaning of comparing distribution functions of quantum observables and organizes the structure of unbounded positive operators into a complete lattice. In the previous investigation, we clarified the structure of canonical preservers of the spectral order relation in the bounded case. In the present paper, we have discussed new results on preservers of the spectral order for unbounded positive operators affiliated with von Neumann algebras. We proved earlier that any spectral automorphisms (bijection preserving the order in both directions) of the set of all positive unbounded operators acting on a Hilbert space is a composition of function calculus with a natural extension of projection lattice automorphism. Our investigation starts with observation that this does not hold if the underlying von Neumann algebras have a non-trivial center. However, we have shown that for any von Neumann algebra the following holds. The spectral automorphism preserves positive multiples of projections if and only if it is a composition of the function calculus given by a strictly increasing bijection of the positive part of the real line and an extension of projection lattice automorphism.

Keywords: spectral order, unbounded operators

Acknowledgements. The work by J. Hamhalter was supported by the "Czech Science Foundation'' (project no. 17-00941S, "Topological and geometrical properties of Banach spaces and operator algebras II''). The work by E.A. Turilova was supported by subsidies allocated to Kazan Federal University for the state assignment in the sphere of scientific activities (no. 1.7629.2017/8.9).

References

1. Kadison R.V. Order properties of bounded self-adjoint operators. Proc. Am. Math. Soc., 1951, vol. 2, pp. 505–510.

2. Olson M.P. The self-adjoint operators of a von Neumann algebra form a conditionally complete lattice. Proc. Am. Math. Soc., 1971, vol. 28, no. 2, pp. 537–544.

3. Bush P., Grabowski M., Lahti P.J. Operational Quantum Physics. Berlin, Heidelberg, Springer, 1995. xi, 232 p. doi: 10.1007/978-3-540-49239-9.

4. Hamhalter J. Quantum Measure Theory. Springer Neth., 2003. viii, 410 p. doi: 10.1007/978-94-017-0119-8.

5. Landsman K. Foundations of Quantum Theory. From Classical Concepts to Operator Algebras. Springer, 2017. xxxvi, 861 p. doi: 10.1007/978-3-319-51777-3.

6. Arveson W. On groups of automorphisms of operator algebras. J. Funct. Anal., 1974, vol. 15, no. 3, pp. 217–243. doi: 10.1016/0022-1236(74)90034-2.

7. Ando T. Majorization, doubly stochastic matrices, and comparison of eigenvalues. Linear Algebra Its Appl., vol. 118, pp. 163–248. doi: 10.1016/0024-3795(89)90580-6.

8. de Groote H.F. On the canonical lattice structure on the effect algebra of a von Neumann algebra. arXiv:math-ph/0410018, 2004, pp. 1–18.

9. Hamhalter J. Spectral order of operators and range projections. J. Math. Anal. Appl., 2007, vol. 331, no. 2, pp. 1122–1134. doi: 10.1016/j.jmaa.2006.09.045.

10. Hamhalter J. Spectral lattices. Int. J. Theor. Phys., 2008, vol. 47, no. 1, pp. 245–251. doi: 10.1007/s10773-007-9464-5.

11. Mitra S.K., Bhimasankaram P., Malik S.B. Matrix Partial Orders, Shorted Operators and Applications. Singapore, World Sci. Publ. Co., 2010. 464 p.

12. Molnar L., Semrl P. Spectral order automorphisms of the spaces of Hilbert space effects and observables. Lett. Math. Phys., 2007, vol. 80, no. 3, pp. 239–255. doi: 10.1007/s11005-007-0160-4.

13. Hamhalter J., Turilova E. Spectral order on aw-algebras and its preservers. Lobachevskii J. Math., 2016, vol. 37, no. 4, pp. 439–448. doi: 10.1134/S1995080216040107.

14. Hamhalter J., Turilova E. Quantum spectral symmetries. Int. J. Theor. Phys., 2017, vol. 56, no. 12, pp. 3807–3818. doi: 10.1007/s10773-017-3312-z.

15. Turilova E. Automorphisms of spectral lattices of unbounded positive operators. Lobachevskii J. Math., 2014, vol. 35, no. 3, pp. 259–263. doi: 10.1134/S1995080214030111.

16. Turilova E. Automorphisms of spectral lattices of positive contractionson von Neumann algebras. Lobachevskii J. Math., 2014, vol. 35, no. 4, pp. 355–359. doi: 10.1134/S1995080214040222.

17. Schumudgen K. Unbounded Self-Adjoint Operators on Hilbert Space. Springer Neth., 2012. xx, 432 p. doi: 10.1007/978-94-007-4753-1.

18. Kadison R.V., Ringrose J.R. Fundamentals of the Theory of Operator Algebras. Vol I: Elementary Theory (Pure and Applied Mathematics). London, Acad. Press, 1983. 398 p.

Received

November 21, 2017

For citation: Hamhalter J., Turilova E.A. Spectral order on unbounded operators and their symmetries. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 2, pp. 293–299.

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