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Keywords:
complex $L_1$-predual; extreme point; Baire function
complex $L_1$-predual; extreme point; Baire function
Summary:
Let $X$ be a complex \mbox {$L_1$-predual}, non-separable in general. We investigate extendability of complex-valued bounded homogeneous Baire-$\alpha $ functions on the set $\mathop {\rm ext} B_{X^*}$ of the extreme points of the dual unit ball $B_{X^*}$ to the whole unit ball $B_{X^*}$. As a corollary we show that, given $\alpha \in [1,\omega _1)$, the intrinsic \mbox {$\alpha $-th} Baire class of $X$ can be identified with the space of bounded homogeneous Baire-$\alpha $ functions on the set $\mathop {\rm ext} B_{X^*}$ when $\mathop {\rm ext} B_{X^*}$ satisfies certain topological assumptions. The paper is intended to be a complex counterpart to the same authors' paper: Baire classes of non-separable $L_1$-preduals (2015). As such it generalizes former work of Lindenstrauss and Wulbert (1969), Jellett (1985), and ourselves (2014), (2015).
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] Argyros, S. A., Godefroy, G., Rosenthal, H. P.: Descriptive set theory and Banach spaces. Handbook of the Geometry of Banach Spaces, Vol. 2 W. B. Johnson et al. North-Holland Amsterdam (2003), 1007-1069. DOI 10.1016/S1874-5849(03)80030-X | MR 1999190 | Zbl 1121.46008
[3] Effros, E. G.: On a class of complex Banach spaces. Ill. J. Math. 18 (1974), 48-59. DOI 10.1215/ijm/1256051348 | MR 0328548 | Zbl 0291.46011
[4] Ellis, A. J., Rao, T. S. S. R. K., Roy, A. K., Uttersrud, U.: Facial characterizations of complex Lindenstrauss spaces. Trans. Am. Math. Soc. 268 (1981), 173-186. DOI 10.1090/S0002-9947-1981-0628453-7 | MR 0628453 | Zbl 0538.46013
[5] Holický, P., Kalenda, O.: Descriptive properties of spaces of measures. Bull. Pol. Acad. Sci., Math. 47 (1999), 37-51. MR 1685676 | Zbl 0929.54026
[6] Hustad, O.: Intersection properties of balls in complex Banach spaces whose duals are {$L_1$} spaces. Acta Math. 132 (1974), 283-313. DOI 10.1007/BF02392118 | MR 0388049 | Zbl 0309.46025
[7] Jellett, F.: On affine extensions of continuous functions defined on the extreme boundary of a Choquet simplex. Q. J. Math., Oxf. II. Ser. 36 (1985), 71-73. DOI 10.1093/qmath/36.1.71 | MR 0780351 | Zbl 0582.46010
[8] Kuratowski, K.: Topology. Vol. I. New edition, revised and augmented Academic Press, New York; PWN-Polish Scientific Publishers, Warsaw (1966). MR 0217751 | Zbl 0158.40901
[9] 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
[10] Lazar, A. J.: The unit ball in conjugate $L_1$ spaces. Duke Math. J. 39 (1972), 1-8. DOI 10.1215/S0012-7094-72-03901-4 | MR 0303242
[11] Lima, A.: Complex Banach spaces whose duals are $L_1$-spaces. Isr. J. Math. 24 (1976), 59-72. DOI 10.1007/BF02761429 | MR 0425584 | Zbl 0334.46014
[12] Lindenstrauss, J., Wulbert, D. E.: On the classification of the Banach spaces whose duals are $L_1$ spaces. J. Funct. Anal. 4 (1969), 332-349. DOI 10.1016/0022-1236(69)90003-2 | MR 0250033
[13] Ludvík, P., Spurný, J.: Baire classes of non-separable $L_1$-preduals. Q. J. Math. 66 (2015), 251-263. DOI 10.1093/qmath/hau007 | MR 3356290
[14] Ludvík, P., Spurný, J.: Baire classes of $L_1$-preduals and $C^*$-algebras. Ill. J. Math. 58 (2014), 97-112. DOI 10.1215/ijm/1427897169 | MR 3331842
[15] Ludvík, P., Spurný, J.: Descriptive properties of elements of biduals of Banach spaces. Stud. Math. 209 (2012), 71-99. DOI 10.4064/sm209-1-6 | MR 2914930
[16] 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, Berlin (2010). MR 2589994 | Zbl 1216.46003
[17] Lusky, W.: Every separable $L_1$-predual is complemented in a $C^*$-algebra. Stud. Math. 160 (2004), 103-116. DOI 10.4064/sm160-2-1 | MR 2033145 | Zbl 1054.46009
[18] Olsen, G. H.: On the classification of complex Lindenstrauss spaces. Math. Scand. 35 (1975), 237-258. DOI 10.7146/math.scand.a-11550 | MR 0367626 | Zbl 0325.46021
[19] Rogers, C. A., Jayne, J. E.: $K$-analytic sets. Analytic Sets. Lectures delivered at the London Mathematical Society Instructional Conference on Analytic Sets held at University College, University of London, 1978. Academic Press London (1980), 1-181. MR 0608794
[20] Roy, A. K.: Convex functions on the dual ball of a complex Lindenstrauss space. J. Lond. Math. Soc., II. Ser. 20 (1979), 529-540. DOI 10.1112/jlms/s2-20.3.529 | MR 0561144 | Zbl 0421.46008
[21] Rudin, W.: Real and Complex Analysis. McGraw-Hill New York (1987). MR 0924157 | Zbl 0925.00005
[22] Talagrand, M.: A new type of affine Borel function. Math. Scand. 54 (1984), 183-188. DOI 10.7146/math.scand.a-12052 | MR 0757461 | Zbl 0562.46005
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