Previous |  Up |  Next

Article

Keywords:
complex Banach spaces; $L_1$-predual; barycentric mapping
Summary:
We provide a complex version of a theorem due to Bednar and Lacey characterizing real $L_1$-preduals. Hence we prove a characterization of complex $L_1$-preduals via a complex barycentric mapping.
References:
[1] Alfsen E.M.: Compact Convex Sets and Boundary Integrals. Ergebnisse der Mathematik und ihrer Grenzgebiete, 57, Springer, New York, 1971. MR 0445271 | Zbl 0209.42601
[2] Bednar J., Lacey H.: Concerning Banach spaces whose duals are abstract L-spaces. Pacific J. Math. 41 (1972), no. 1, 13–24. DOI 10.2140/pjm.1972.41.13 | MR 0308747
[3] Effros E.G.: On a class of complex Banach spaces. Illinois J. Math. 18 (1974), 48–59. MR 0328548
[4] Ellis A.J., Rao T.S.S.R.K., Roy A.K., Uttersrud U.: Facial characterizations of complex Lindenstrauss spaces. Trans. Amer. Math. Soc. 268 (1981), no. 1, 173–186. DOI 10.1090/S0002-9947-1981-0628453-7 | MR 0628453
[5] Fabian M., Habala P., Hájek P., Montesinos V., Zizler V.: Banach Space Theory. The Basis for Linear and Nonlinear Analysis, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, 2011. MR 2766381 | Zbl 1229.46001
[6] Hustad O.: Intersection properties of balls in complex Banach spaces whose duals are ${L_1}$-spaces. Acta Math. 132 (1974), no. 1, 283–313. DOI 10.1007/BF02392118 | MR 0388049
[7] Krause U.: Der Satz von Choquet als ein abstrakter Spektralsatz und vice versa. Math. Ann. 184 (1970), no. 4, 275–296. DOI 10.1007/BF01350856 | MR 1513280
[8] Lacey H.E.: The Isometric Theory of Classical Banach Spaces. Die Grundlehren der mathematischen Wissenschaften, 208, Springer, New York, 1974. MR 0493279 | Zbl 0285.46024
[9] Lima A.: Complex Banach spaces whose duals are $L_1$-spaces. Israel J. Math. 24 (1976), no. 1, 59–72. DOI 10.1007/BF02761429 | MR 0425584
[10] Ludvík P., Spurný J.: Baire classes of complex $L_1$-preduals. Czechoslovak Math. J. 65(140) (2015), no. 3, 659–676. DOI 10.1007/s10587-015-0201-6 | MR 3407598
[11] Ludvík P., Spurný J.: Baire classes of $L_1$-preduals and $C^*$-algebras. Complex Banach spaces whose duals are $L_1$-spaces, Illinois J. Math. 58 (2014), no. 1, 97–112. MR 3331842
[12] Ludvík P., Spurný J.: Baire classes of nonseparable $L_1$-preduals. Q.J. Math. 66 (2015), no. 1, 251–263. DOI 10.1093/qmath/hau007
[13] Ludvík P., Spurný J.: Descriptive properties of elements of biduals of Banach spaces. Studia Math. 209 (2012), no. 1, 71–99. DOI 10.4064/sm209-1-6 | MR 2914930
[14] Lukeš J., Malý J., Netuka I., Spurný J.: Integral Representation Theory. Applications to Convexity, Banach Spaces and Potential Theory, de Gruyter Studies in Mathematics, 35, Walter de Gruyter & Co., Berlin, 2010. MR 2589994
[15] Lusky W.: Every separable $L_1$-predual is complemented in a $C^*$-algebra. Studia Math. 160 (2004), no. 2, 103–116. DOI 10.4064/sm160-2-1 | MR 2033145
[16] Olsen G.H.: On the classification of complex Lindenstrauss spaces. Mathematica Scand. 35 (1974), 237–258. MR 0367626
[17] Roy A.K.: Convex functions on the dual ball of a complex Lindenstrauss space. J. London Math. Soc. (2) 20 (1979), no. 3, 529–540. DOI 10.1112/jlms/s2-20.3.529 | MR 0561144
[18] Spurný J.: Borel sets and functions in topological spaces. Acta Math. Hungar. 129 (2010), no. (1-2), 47–69. DOI 10.1007/s10474-010-9223-6 | MR 2725834
Partner of
EuDML logo