Complementation of Jordan triples in von Neumann algebras

Chu, Cho-Ho; Iochum, Bruno

doi:10.1090/S0002-9939-1990-0990418-4

Complementation of Jordan triples in von Neumann algebras
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by Cho-Ho Chu and Bruno Iochum

Proc. Amer. Math. Soc. 108 (1990), 19-24

DOI: https://doi.org/10.1090/S0002-9939-1990-0990418-4

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Abstract:

We show that the predual of a JBW*-triple is complemented in the predual of a von Neumann algebra. Hence a quotientof a JB*-triple does not contain ${l_1}$ if and only if its dual enjoys the Radon-Nikodym property. We also show that a JB*-triple either contains ${c_0}$ or is reflexive.

References

C. A. Akemann, P. G. Dodds, and J. L. B. Gamlen, Weak compactness in the dual space of $C^{\ast }$-algebra, J. Functional Analysis 10 (1972), 446–450. MR 0344898, DOI 10.1016/0022-1236(72)90040-7
Tom Barton and Gilles Godefroy, Remarks on the predual of a $\textrm {JB}^\ast$-triple, J. London Math. Soc. (2) 34 (1986), no. 2, 300–304. MR 856513, DOI 10.1112/jlms/s2-34.2.300
T. Barton and Richard M. Timoney, Weak$^\ast$-continuity of Jordan triple products and its applications, Math. Scand. 59 (1986), no. 2, 177–191. MR 884654, DOI 10.7146/math.scand.a-12160
Cho-Ho Chu and Bruno Iochum, Weakly compact operators on Jordan triples, Math. Ann. 281 (1988), no. 3, 451–458. MR 954152, DOI 10.1007/BF01457156
Edward G. Effros and Erling Størmer, Positive projections and Jordan structure in operator algebras, Math. Scand. 45 (1979), no. 1, 127–138. MR 567438, DOI 10.7146/math.scand.a-11830
Yaakov Friedman and Bernard Russo, Algèbres d’opérateurs sans ordre: solution du problème du projecteur contractif, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 9, 393–396 (French, with English summary). MR 703905
N. Ghoussoub, G. Godefroy, B. Maurey, and W. Schachermayer, Some topological and geometrical structures in Banach spaces, Mem. Amer. Math. Soc. 70 (1987), no. 378, iv+116. MR 912637, DOI 10.1090/memo/0378
Nassif Ghoussoub and Elias Saab, On the weak Radon-Nikodým property, Proc. Amer. Math. Soc. 81 (1981), no. 1, 81–84. MR 589141, DOI 10.1090/S0002-9939-1981-0589141-4

Jordan operator algebras

Günther Horn, Classification of JBW$^*$-triples of type $\textrm {I}$, Math. Z. 196 (1987), no. 2, 271–291. MR 910832, DOI 10.1007/BF01163661
Günther Horn, Characterization of the predual and ideal structure of a $\textrm {JBW}^*$-triple, Math. Scand. 61 (1987), no. 1, 117–133. MR 929400, DOI 10.7146/math.scand.a-12194
G. Horn and E. Neher, Classification of continuous $JBW^*$-triples, Trans. Amer. Math. Soc. 306 (1988), no. 2, 553–578. MR 933306, DOI 10.1090/S0002-9947-1988-0933306-7
Bruno Iochum, Cônes autopolaires et algèbres de Jordan, Lecture Notes in Mathematics, vol. 1049, Springer-Verlag, Berlin, 1984 (French). MR 764767, DOI 10.1007/BFb0071358
Liliana Janicka, Some measure-theoretical characterization of Banach spaces not containing $l_{1}$, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 7-8, 561–565 (1980) (English, with Russian summary). MR 581552
Wilhelm Kaup, A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z. 183 (1983), no. 4, 503–529. MR 710768, DOI 10.1007/BF01173928
Kazimierz Musiał and Czesław Ryll-Nardzewski, Liftings of vector measures and their applications to RNP and WRNP, Vector space measures and applications (Proc. Conf., Univ. Dublin, Dublin, 1977) Lecture Notes in Math., vol. 645, Springer, Berlin-New York, 1978, pp. 162–171. MR 502438
Constantin P. Niculescu, A note on weakly compact operators on $C^{\ast }$-algebras, Rev. Roumaine Math. Pures Appl. 25 (1980), no. 4, 631–634. MR 577053
Harald Upmeier, Symmetric Banach manifolds and Jordan $C^\ast$-algebras, North-Holland Mathematics Studies, vol. 104, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 96. MR 776786

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Complementation of Jordan triples in von Neumann algebras
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Abstract: