Geodesic
Reversible AJW-algebras
Vladikavkazskij matematičeskij žurnal, Tome 18 (2016) no. 3, pp. 15-21 
Sh. A. Ayupov; F. N. Arzikulov. Reversible AJW-algebras. Vladikavkazskij matematičeskij žurnal, Tome 18 (2016) no. 3, pp. 15-21. http://geodesic.mathdoc.fr/item/VMJ_2016_18_3_a1/
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     author = {Sh. A. Ayupov and F. N. Arzikulov},
     title = {Reversible {AJW-algebras}},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The main result states that every special AJW-algebra can be decomposed into the direct sum of totally irreversible and reversible subalgebras. In turn, every reversible special AJW-algebra decomposes into a direct sum of two subalgebras, one of which has purely real enveloping real von Neumann algebra, and the second one contains an ideal, whose complexification is a C$^*$-algebra and the annihilator of this complexification in the enveloping $C^*$-algebra of this subalgebra is equal to zero.

[1] Albeverio S., Ayupov Sh. A., Abduvaitov A. H., “On Real $AW^*$-algebras”, Methods of Functional Analysis and Topology, 11:2 (2005), 99–112 | MR | Zbl

[2] Albeverio S., Ayupov Sh. A., Abduvaitov A. H., “On the coincidence of types of a real $AW^*$-algebra and its comlexification”, News of Russian Acad. of Sci. Math. ser., 68 (2004), 3–12 (Russian) | MR | Zbl

[3] Arzikulov F. N., “On abstract $JW$-algebras”, Sib. Math. J., 39:1 (1998), 20–27 | DOI | MR | Zbl

[4] Arzikulov F. N., “On an analog of the Peirce decomposition”, Sib. Math. J., 40:3 (1999), 413–419 | DOI | MR | Zbl

[5] Arzikulov F. N., “AJW-algebras of type I and their classification”, Sib. Adv. Math., 8:2 (1998), 30–48 | MR | Zbl

[6] Ayupov Sh. A., “Traces on JW-algebras and enveloping $W^*$-algebras”, Math. Z., 194 (1987), 15–23 | DOI | MR | Zbl

[7] Ayupov Sh., Classification and representation of ordered Jordan algebras, Fan, Tashkent, 1986 (Russian) | MR | Zbl

[8] Chilin V. I., “Partial ordered Baer involutive algebras”, Itogi nauki i tehkniki. Sovrem. Problemy Matematiki, 27, VINITI, 1985, 99–128 (Russian) | MR | Zbl

[9] Hanche-Olcen H., Størmer E., Jordan Operator Algebras, Pitman Publ. Inc., Boston etc., 1984 | MR

[10] Kusraev A. G., “On the structure of AJW-algebras of type $I_2$”, Sib. Math. J., 40:4 (1999), 905–917 | DOI | MR | Zbl

[11] Li B., Real operator algebras, Institute of Mathematics, Acad. Sinica Beijing, China, 2002 | MR

[12] Sakai S., $C^*$-algebras and $W^*$-algebras, Springer, 1971 | MR

[13] Størmer E., “Jordan algebras of type I”, Acta. Math., 115 (1966), 165–184 | DOI | MR

[14] Topping D. M., Jordan algebras of self-adjoint operators, Mem. Am. Math. Soc., 53, 1965 | MR | Zbl