Abstract
Self-adjoint operators represent bounded observables in mathematical foundations of quantum mechanics. The set of all self-adjoint operators can be equipped with several operations and relations having important interpretations in physics. Automorphisms with respect to these relations or operations are called symmetries. Many of them turn out to be real-linear up to a translation. We present a unified approach to the description of the general form of such symmetries based on adjacency preserving maps. We consider also symmetries defined on the set of all positive operators or on the set of all positive invertible operators. In particular, we will see that the structural result for adjacency preserving maps on the set of all positive invertible operators differs a lot from its counterpart on the set of all selfadjoint operators.
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References
An R., Hou J.: Additivity of Jordan multiplicative maps on Jordan operator algebras. Taiwan. J. Math. 10, 45–64 (2006)
Ando T.: Topics on operator inequalities, Division of Applied Math. Research Institute of Applied Electricity, Hokkaido University, Sapporo (1978)
Cassinelli G., De Vito E., Lahti P., Levrero A.: The Theory of Symmetry Actions in Quantum Mechanics, Lecture Notes in Physics, vol. 654. Springer, New York (2004)
Havlicek H., Huang W.-l.: Diameter preserving surjections in the geometry of matrices. Linear Algebra Appl. 429, 376–386 (2008)
Havlicek H., Šemrl P.: From geometry to invertibility preservers. Studia Math. 174, 99–109 (2006)
Huang W.-l., Höfer R., Wan Z.-X.: Adjacency preserving mappings of symmetric and Hermitian matrices. Aequationes Math. 67, 32–139 (2004)
Huang W.-l., Šemrl P.: Adjacency preserving maps on hermitian matrices. Canad. J. Math. 60, 1050–1066 (2008)
Kadison R.V.: A generalized Schwarz inequality and algebraic invariants for operator algebras. Ann. Math. 56, 494–503 (1952)
Kobal D.: Bijections preserving invertibility of differences of matrices on H n . Acta Math. Sinica (Engl. Ser.) 24, 1651–1654 (2008)
Molnár L.: On some automorphisms of the set of effects on Hilbert space. Lett. Math. Phys. 51, 37–45 (2000)
Molnár L.: Order-automorphisms of the set of bounded observables. J. Math. Phys. 42, 5904–5909 (2001)
Molnár L.: Non-linear Jordan triple automorphisms of sets of self-adjoint matrices and operators. Studia Math. 173, 39–48 (2006)
Molnár L.: Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces. Springer, Berlin (2007)
Molnár L.: Maps preserving the geometric mean of positive operators. Proc. Am. Math. Soc 137, 1763–1770 (2009)
Molnár L., Šemrl P.: Nonlinear commutativity preserving maps on self-adjoint operators. Quart. J. Math 56, 589–595 (2005)
Murphy G.J.: C*-algebras and operator theory. Academic Press, San Diego (1990)
Radjavi H., Šemrl P.: A short proof of Hua’s fundamental theorem of the geometry of hermitian matrices. Expo. Math. 21, 83–93 (2003)
Rothaus O.S.: Order isomorphisms of cones. Proc. Am. Math. Soc. 17, 1284–1288 (1966)
Šemrl P.: Comparability preserving maps on bounded observables. Int. Equ. Oper. Theory 62, 441–454 (2008)
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This work was partially supported by a grant from the Ministry of Science of Slovenia.
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Šemrl, P. Symmetries on Bounded Observables: A Unified Approach Based on Adjacency Preserving Maps. Integr. Equ. Oper. Theory 72, 7–66 (2012). https://doi.org/10.1007/s00020-011-1917-9
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DOI: https://doi.org/10.1007/s00020-011-1917-9