×

Banach-power-associative algebras and J-B algebras. (English) Zbl 0587.17012

The paper closes the gap in the Jordan-von Neumann-Wigner axiomatic approach to quantum theory by proving that, even in the presence of a norm topology in the family of bounded observables of a quantum system, some elementary axioms imply the Jordan structure in this family. Namely it is shown that the set of observables of a quantum system, stable under the operations of the linear combination and the square, and complete with respect to a compatible norm topology, is a Jordan-Banach algebra, whose structure is by now very well described.
Reviewer: W.Guz

MSC:

17C65 Jordan structures on Banach spaces and algebras
17A05 Power-associative rings
46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
17C50 Jordan structures associated with other structures
PDF BibTeX XML Cite
Full Text: Numdam EuDML

References:

[1] A.A. Albert , On the power-associativity of rings , Summa Brasil, Math. , t. 2 , 1948 , p. 21 - 32 . MR 26044 | Zbl 0039.26403 · Zbl 0039.26403
[2] A.A. Albert , A theory of power-associative commutative algebras , Trans. Amer. Math. Soc. , t. 69 , 1950 , p. 503 - 527 . MR 38959 | Zbl 0039.26501 · Zbl 0039.26501
[3] E.M. Alfsen , Compact convex sets and boundary integrals , Ergebnisse der Math. , t. 57 , Springer-Verlag: Berlin , 1971 . MR 445271 | Zbl 0209.42601 · Zbl 0209.42601
[4] E.M. Alfsen and F.W. Shultz , Non commutative spectral theory for affine function spaces on convex sets , Mem. Amer. Math. Soc. , t. 172 , 1976 . MR 412822 | Zbl 0337.46013 · Zbl 0337.46013
[5] E.M. Alfsen and F.W. Shultz , On non commutative spectral theory and Jordan algebras , Proc. London Math. Soc. , t. 38 , 1979 , p. 497 - 516 . MR 532984 | Zbl 0404.46028 · Zbl 0404.46028
[6] E.M. Alfsen , F.W. Shultz and E. Størmer , A Guelfand-Neumark theorem for Jordan algebras , Adv. Math. , t. 28 , 1978 , p. 11 - 56 . MR 482210 | Zbl 0397.46065 · Zbl 0397.46065
[7] G.G. Emch , Algebraic methods in statistical mechanics and quantum field theory , Wiley-Interscience : New York , 1972 . Zbl 0235.46085 · Zbl 0235.46085
[8] W. Guz , Conditional probability in quantum axiomatic , Ann. Inst. H. Poincaré , 1980 , p. 63 - 119 . Numdam | MR 593025 | Zbl 0454.03032 · Zbl 0454.03032
[9] R. Haag and D. Kastler , An algebraic approach to quantum field theory , J. Math. Phys. , t. 5 , 1964 , p. 848 - 861 . MR 165864 | Zbl 0139.46003 · Zbl 0139.46003
[10] H. Hanche-Olsen , A note on the bidual of a Jordan-Banach-algebra , Math. Z. , t. 175 , 1980 , p. 29 - 31 . MR 595629 | Zbl 0424.46036 · Zbl 0424.46036
[11] H. Hanche-Olsen and E. Størmer , Jordan operator algebras , Pitman , Boston , 1984 . MR 755003 | Zbl 0561.46031 · Zbl 0561.46031
[12] B. Iochum , Cônes autopolaires et algèbres de Jordan , Lectures Notes in Math. , 1049 Springer-Verlag , Berlin , 1984 . MR 764767 | Zbl 0556.46040 · Zbl 0556.46040
[13] B. Iochum and F.W. Shultz , Normal state spaces of Jordan and von Neumann algebra , J. Functional Anal. , t. 50 , 1983 , p. 317 - 328 . MR 695418 | Zbl 0507.46055 · Zbl 0507.46055
[14] P. Jordan , J. Von Neumann and E. Wigner , On an algebraic generalization of the quantum mechanical formalism , Ann. of Math. , t. 35 , 1934 , p. 29 - 64 . MR 1503141 | Zbl 0008.42103 | JFM 60.0902.02 · Zbl 0008.42103
[15] R.V. Kadison , A representation theory for commutative topological algebra , Mem. Amer. Math. Soc. , t. 7 , 1951 . MR 44040 | Zbl 0042.34801 · Zbl 0042.34801
[16] R.V. Kadison , A generalized Schwarz inequality and algebraic invariants for operator algebras , Ann. Math. , t. 56 , 1952 , p. 494 - 503 . MR 51442 | Zbl 0047.35703 · Zbl 0047.35703
[17] L.A. Kokoris , Simple power associative algebras of degree two , Ann. of Math. , t. 64 , 1956 , p. 544 - 550 . MR 81273 | Zbl 0072.26202 · Zbl 0072.26202
[18] D.B. Lowdenslager , On postulates for general quantum mechanics , Proc. Amer. Math. Soc. , t. 8 , 1957 , p. 88 - 91 . MR 84741 | Zbl 0079.13003 · Zbl 0079.13003
[19] J. Von Neumann , On an algebraic generalization of the quantum mechanical formalism (Part I) , Math. Sbornic , t. 1 , 1936 , p. 415 - 484 . Zbl 0015.24505 · Zbl 0015.24505
[20] S. Sakai , C* and W* algebras , Ergebnisse der Math. , t. 60 , Springer-Verlag , Berlin , 1971 . MR 442701 | Zbl 0219.46042 · Zbl 0219.46042
[21] H.H. Schaeffer , Banach lattices and positive operators , Springer-Verlag , Berlin , 1974 . Zbl 0296.47023 · Zbl 0296.47023
[22] R.D. Schafer , An introduction to non associative algebras , Academic Press , New York , 1966 . MR 210757 | Zbl 0145.25601 · Zbl 0145.25601
[23] S. Sherman , On Segal’s postulates for general quantum mechanics , Ann. of Math. , t. 64 , 1956 , p. 593 - 601 . MR 82887 | Zbl 0075.21802 · Zbl 0075.21802
[24] F.W. Shultz , On normed Jordan algebras which are Banach dual spaces . J. Functional Anal. , t. 31 , 1979 , p. 360 - 376 . MR 531138 | Zbl 0421.46043 · Zbl 0421.46043
[25] I.E. Segal , Postulates for a general quantum mechanics , Ann. of Math. , t. 48 , 1947 , p. 930 - 948 . MR 22652 | Zbl 0034.06602 · Zbl 0034.06602
[26] R.R. Smith , On non-unital Jordan-Banach algebras , Math. Proc. Camb. Phil. Soc.. t. 82 , 1977 , p. 375 - 380 . MR 467328 | Zbl 0372.46045 · Zbl 0372.46045
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.