×

zbMATH — the first resource for mathematics

Quantum spectral symmetries. (English) Zbl 1387.81232
Summary: Quantum symmetries of spectral lattices are studied. Basic properties of spectral order on \(AW^\ast\)-algebras are summarized. Connection between projection and spectral automorphisms is clarified by showing that, under mild conditions, any spectral automorphism is a composition of function calculus and Jordan \(\ast\)-automorphism. Complete description of quantum spectral symmetries on Type I and Type II \(AW^\ast\)-factors are completely described.

MSC:
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
46L05 General theory of \(C^*\)-algebras
46L40 Automorphisms of selfadjoint operator algebras
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Akemann, C; Weaver, N, Minimal upper bounds of commuting operators, Proc. Amer. Math. Soc., 124, 11, (1996) · Zbl 0863.46037
[2] Ando, T, Majorization, doubly stochastic matrices, and comparison of eigenvalues, Linear Algebra Appl., 118, 163-248, (1989) · Zbl 0673.15011
[3] Arveson, W, On groups of automorphisms of operator algebras, J. Funct. Anal., 15, 217-243, (1974) · Zbl 0296.46064
[4] Barvinek, J., Hamhalter, J.: Linear algebraic proof of Wigner Theorem and its consequences, Mathematica Slovaca, to appear · Zbl 1424.81005
[5] Berberian, S. K.: Bear ∗ ring, Grundlehren der mathematischen Wissenschaften, pp. 195. Springer (1972) · Zbl 0043.11501
[6] Bush, P., Grabowski, M., Lahti, P.: Operational Quantum Physics. Lecture Notes in Physics. Springer Verlag (1995) · Zbl 0215.20504
[7] Dye, HA, On the geometry of projections in certain operator algebras, Ann. Math., 61, 73-89, (1955) · Zbl 0064.11002
[8] de Groote, H. F: On the canonical lattice structure on the effect algebra of a von Neumann algebra. arXiv:math-ph/0410018v16 (2004)
[9] Hamhalter, J.: Quantum Measure Theory. Kluwer Academic (2003) · Zbl 1038.81003
[10] Hamhalter, J, Spectral order of operators and range projections, J. Math. Anal. Appl., 331, 1122-1134, (2007) · Zbl 1120.46040
[11] Hamhalter, J, Spectral lattices, Int. J. Theor. Phys., 47, 245-251, (2008) · Zbl 1145.81009
[12] Hamhalter, J, Dye theorem and Gleason theorem for AW∗-algebras, J. Math. Anal. Appl., 422, 1103-1115, (2015) · Zbl 1320.46041
[13] Hamhalter, J; Turilova, E, Spectral order on AW∗-algebras and its preservers, Lobachevskii J. Math., 37, 439-448, (2016) · Zbl 1420.46042
[14] Kadison, RV, Order properties of bounded self-adjoint operators, Proc. Amer. Math. Soc, 2, 505-510, (1951) · Zbl 0043.11501
[15] Kadison, R. V., Ringrose, J. R.: Fundamentals of the Theory of Operator Algebras, vol I. Acedemic Press (1983) · Zbl 0518.46046
[16] Kaplansky, I, Projections in Banach algebras, Ann. Math., 53, 2, (1951) · Zbl 0042.12402
[17] Landsman, K.: Bohrification from classical concepts to commutative operator algebras. Springer Verlag, to appear · Zbl 1380.81028
[18] Lindenhovous, B.: \(C\)(\(A\)), PhD. Thesis. Radboud University, Netherlands (2016) · Zbl 0863.46037
[19] Mitra, S. K., Bhimasankaram, P., Malik, S. B.: Matrix Partial Orders, Shorted Operators and Applications. World Scientific Publishing Co., Singapore (2010) · Zbl 1203.15023
[20] Molnar, L; Šemrl, P, Spectral order automorphisms of the spaces of Hilbert space effects and observables, Lett. Math. Phys., 80, 239-255, (2007) · Zbl 1138.47057
[21] Ogasawara, T, A theorem on operator algebras, J. Sci. Hiroshima Univ, 18, 307-309, (1955) · Zbl 0064.36704
[22] Olson, MP, The self-adjoint operators of a von Neumann algebra form a conditionally complete lattice, Proc. Amer. Math. Soc., 28, 537-544, (1971) · Zbl 0215.20504
[23] Pedersen, G. K.: C \^{∗}-algebras and their Automorphism Groups. Acedemic Press (1979) · Zbl 0416.46043
[24] Saito, K., Wright, J. D. M.: Monotone Complete C*-algebras and Generic Dynamics, Springer Monographs in Mathematics. Springer-Verlag, London (2015) · Zbl 1382.46003
[25] Sherman, S, Order in operator algebras, Amer. J. Math, 73, 227-232, (1951) · Zbl 0042.35001
[26] Stratila, S., Szido, L.: Operator Algebras, Part II, Monogafii Matematice, pp. 54. University of Timisoara (1995) · Zbl 1326.46050
[27] Turilova, E, Automorphisms of spectral lattices of unbounded positive operators, Lobachevskii J. Math., 35, 258-262, (2014) · Zbl 1331.47029
[28] Turilova, E, Automorphisms of spectral lattices of positive contractions on von Neumann algebras, Lobachevskii J. Math., 35, 354-358, (2014) · Zbl 1326.46050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.