Geodesic
Geometry of the state space of modular Jordan algebras
Izvestiya. Mathematics , Tome 43 (1994) no. 3, pp. 581-592.

Voir la notice de l'article provenant de la source Math-Net.Ru

In the class of compact convex sets the normal state spaces of modular Jordan algebras are characterized. To wit,the state spaces are precisely those sets that are modular, spectral, and symmetric. 
@article{IM2_1994_43_3_a9,
     author = {Sh. A. Ayupov and N. D. Yadgorov},
     title = {Geometry of the state space of modular {Jordan} algebras},
     journal = {Izvestiya. Mathematics },
     pages = {581--592},
     publisher = {mathdoc},
     volume = {43},
     number = {3},
     year = {1994},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1994_43_3_a9/}
}
TY  - JOUR
AU  - Sh. A. Ayupov
AU  - N. D. Yadgorov
TI  - Geometry of the state space of modular Jordan algebras
JO  - Izvestiya. Mathematics 
PY  - 1994
SP  - 581
EP  - 592
VL  - 43
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1994_43_3_a9/
LA  - en
ID  - IM2_1994_43_3_a9
ER  - 
%0 Journal Article
%A Sh. A. Ayupov
%A N. D. Yadgorov
%T Geometry of the state space of modular Jordan algebras
%J Izvestiya. Mathematics 
%D 1994
%P 581-592
%V 43
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1994_43_3_a9/
%G en
%F IM2_1994_43_3_a9
Sh. A. Ayupov; N. D. Yadgorov. Geometry of the state space of modular Jordan algebras. Izvestiya. Mathematics , Tome 43 (1994) no. 3, pp. 581-592. http://geodesic.mathdoc.fr/item/IM2_1994_43_3_a9/

[1] Jordan P., von Neumann J., Wigner E., “On an algebraic generalization of the quantum mechanical formalism”, Ann. of Math., 35 (1934), 29–64 | DOI | MR | Zbl

[2] Sigal A., Matematicheskie problemy relyativistskoi fiziki, Mir, M., 1968 | MR

[3] Emkh Zh., Algebraicheskie metody v statisticheskoi mekhanike i kvantovoi teorii polya, Mir, M., 1976

[4] Bogolyubov N. N., Logunov A. A., Todorov N. T., Osnovy aksiomaticheskogo podkhoda v kvantovoi teorii polya, Nauka, M., 1969 | MR

[5] Bratteli U., Robinson D., Operatornye algebry i kvantovaya statisticheskaya mekhanika, Mir, M., 1982 | MR | Zbl

[6] Birkhoff G., von Neumann J., “On the logic of quantum mechanics”, Ann. Math., 37 (1936), 823–843 | DOI | MR | Zbl

[7] Varadarajan V. S., Geometry of quantum theory, Springer, Berlin, 1985 | MR | Zbl

[8] Makki Dzh., Lektsii po matematicheskim osnovam kvantovoi mekhaniki, Mir, M., 1965 | MR

[9] Foundations of quantum mechanics and ordered linear spaces, Lecture Notes in Physics, 29, 1974

[10] Ladwig G., Foundations of quantum mechanics, Springer, Berlin, 1983

[11] Kholevo A. C., Veroyatnostnye i statisticheskie aspekty kvantovoi teorii, Nauka, M., 1980 | MR | Zbl

[12] Holevo A. S., “Statistical definition of observable and the structure of statistical models”, Reports of Math. Phys., 22 (1985), 385–407 | DOI | MR | Zbl

[13] Alfsen E. M., Shultz F. W., “Non commutative spectral theory for affine function spaces on convex sets”, Memoirs Amer. Math. Soc., 72, Providence R.I., 1976, 1–120 | MR

[14] Bunce L. J., Wright J. D. M., “Quantum logics and convex geometry”, Commun. Math. Phys., 101 (1985), 87–96 | DOI | MR | Zbl

[15] Chu C. H., “On convexity theory and $C^*$-algebras”, Proc. London Math. Soc., 31 (1975), 257–288 | DOI | MR | Zbl

[16] Alfsen E. M., Shultz F. W., State spaces of Jordan algebras, preprint, Oslo, 1976 | MR

[17] Alfsen E. M., Shultz F. W., “On non-commutative spectral theory and Jordan algebras”, Proc. London Math. Soc., 38 (1979), 497–516 | DOI | MR | Zbl

[18] Alfsen E. M., Shultz F. W., “State Spaces of Jordan algebras”, Acta Math., 40:3,4 (1978), 155–190 | DOI | MR

[19] Alfsen E. M., Hanche-Olsen H., Shultz F. W., “State spaces of $C^{*} $-algebras”, Acta Math., 144:3,4 (1980), 267–305 | DOI | MR | Zbl

[20] Araki H., “On the characterization of the state space of quantum mechanics”, Commun. Math. Physics, 75 (1980), 1–25 | DOI | MR

[21] Iochum B., Shultz F. W., “Normal state spaces of Jordan and von Neumann algebras”, J. of Func. Anal., 50 (1983), 317–328 | DOI | MR | Zbl

[22] Ajupov Sh. A., Iochum B., Yadgorov N. Dz., Symmetry versus facial nomogeneity for selfdual cones, preprint, Centre de Physique Theorique Marseille, 1989

[23] Ayupov Sh. A., Iokum V., Yadgorov N. Zh., “Geometriya prostranstv sostoyanii konechnomernykh iordanovykh algebr”, Izv. AN UzSSR. Ser. fiz.-mat. nauk, 1990, no. 3, 19–22 | MR | Zbl

[24] Alfsen E. M., Shultz F. W., Stormer E., “A Gelfand–Neumark theorem for Jordan algebras”, Advances in Math., 28 (1978), 11–56 | DOI | MR | Zbl

[25] Shultz F. W., “On normed Jordan algebras which are Banach dual spaces”, J. of Func. Anal., 31 (1979), 360–376 | DOI | MR | Zbl

[26] Hanche-Olsen H., Stormer E., Jordan Operator algebras, Pitman. Ltd, London, 1984 | MR | Zbl

[27] Ayupov Sh. A., “Iordanovy operatornye algebry”, Sovremennye problemy matematiki. Noveishie dostizheniya. Itogi nauki i tekhniki, 27, VINITI AN SSSR, M., 1985, 67–98 | MR

[28] Alfsen E. M., Compact convex sets and boundary integrals, Springer, Berlin, 1971, 57 | MR | Zbl

[29] Chu C. H., Wright J. D. M., “A theory of types for convex sets and ordered Banach spaces”, Proc. London. Math. Soc., 36 (1978), 497–517 | DOI | MR

[30] Lyubich Yu. I., “Lineinyi funktsionalnyi analiz”, Sovremennye problemy matematiki. Fundamentalnye napravleniya. Itogi nauki i tekhniki, 19, VINITI AN SSSR, M., 1988, 5–305 | MR

[31] Shefer X., Topologicheskie vektornye prostranstva, Mir, M., 1971 | MR