Facial characterizations of complex Lindenstrauss spaces

Ellis, A. J.; Rao, T. S. S. R. K.; Roy, A. K.; Uttersrud, U.

doi:10.1090/S0002-9947-1981-0628453-7

Facial characterizations of complex Lindenstrauss spaces
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by A. J. Ellis, T. S. S. R. K. Rao, A. K. Roy and U. Uttersrud PDF

Trans. Amer. Math. Soc. 268 (1981), 173-186 Request permission

Abstract:

We characterize complex Banach spaces $A$ whose Banach dual spaces are ${L^1}(\mu )$ spaces in terms of $L$-ideals generated by certain extremal subsets of the closed unit ball $K$ of ${A^{\ast }}$. Our treatment covers the case of spaces $A$ containing constant functions and also spaces not containing constants. Separable spaces are characterized in terms of ${w^{\ast }}$-compact sets of extreme points of $K$, whereas the nonseparable spaces necessitate usage of the ${w^{\ast }}$-closed faces of $K$. Our results represent natural extensions of known characterizations of Choquet simplexes. We obtain also a characterization of complex Lindenstrauss spaces in terms of boundary annihilating measures, and this leads to a characterization of the closed subalgebras of ${C_{\mathbf {C}}}(X)$ which are complex Lindenstrauss spaces.

References

Erik M. Alfsen, Compact convex sets and boundary integrals, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 57, Springer-Verlag, New York-Heidelberg, 1971. MR 0445271
Erik M. Alfsen and Edward G. Effros, Structure in real Banach spaces. I, II, Ann. of Math. (2) 96 (1972), 98–128; ibid. (2) 96 (1972), 129–173. MR 352946, DOI 10.2307/1970895
Errett Bishop and R. R. Phelps, The support functionals of a convex set, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 27–35. MR 0154092
Eggert Briem, Peak sets for the real part of a function algebra, Proc. Amer. Math. Soc. 85 (1982), no. 1, 77–78. MR 647902, DOI 10.1090/S0002-9939-1982-0647902-8
Eggert Briem, A characterization of simplexes by parallel faces, Bull. London Math. Soc. 12 (1980), no. 1, 55–59. MR 565485, DOI 10.1112/blms/12.1.55
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
Edward G. Effros, On a class of real Banach spaces, Israel J. Math. 9 (1971), 430–458. MR 296658, DOI 10.1007/BF02771459
Edward G. Effros, On a class of complex Banach spaces, Illinois J. Math. 18 (1974), 48–59. MR 328548
A. J. Ellis, A facial characterization of Choquet simplexes, Bull. London Math. Soc. 9 (1977), no. 3, 326–327. MR 458119, DOI 10.1112/blms/9.3.326
A. J. Ellis and A. K. Roy, Dilated sets and characterizations of simplexes, Invent. Math. 56 (1980), no. 2, 101–108. MR 558861, DOI 10.1007/BF01392544
Richard Fuhr and R. R. Phelps, Uniqueness of complex representing measures on the Choquet boundary, J. Functional Analysis 14 (1973), 1–27. MR 0361741, DOI 10.1016/0022-1236(73)90027-x
Bent Hirsberg, A measure theoretic characterization of parallel and split faces and their connections with function spaces and algebras, Various Publications Series, No. 16, Aarhus Universitet, Matematisk Institut, Aarhus, 1970. MR 0275105
Bent Hirsberg, $M$-ideals in complex function spaces and algebras, Israel J. Math. 12 (1972), 133–146. MR 315451, DOI 10.1007/BF02764658
B. Hirsberg and A. J. Lazar, Complex Lindenstrauss spaces with extreme points, Trans. Amer. Math. Soc. 186 (1973), 141–150. MR 333671, DOI 10.1090/S0002-9947-1973-0333671-7
Ka Sing Lau, The dual ball of a Lindenstrauss space, Math. Scand. 33 (1973), 323–337 (1974). MR 344854, DOI 10.7146/math.scand.a-11494
A. J. Lazar and J. Lindenstrauss, Banach spaces whose duals are $L_{1}$ spaces and their representing matrices, Acta Math. 126 (1971), 165–193. MR 291771, DOI 10.1007/BF02392030
Åsvald Lima, Intersection properties of balls and subspaces in Banach spaces, Trans. Amer. Math. Soc. 227 (1977), 1–62. MR 430747, DOI 10.1090/S0002-9947-1977-0430747-4
Niels Jørgen Nielsen and Gunnar Hans Olsen, Complex preduals of $L_{1}$ and subspaces of $l^{n}_{\infty }(C)$, Math. Scand. 40 (1977), no. 2, 271–287. MR 454597, DOI 10.7146/math.scand.a-11694
Gunnar Hans Olsen, On the classification of complex Lindenstrauss spaces, Math. Scand. 35 (1974), 237–258. MR 367626, DOI 10.7146/math.scand.a-11550
R. R. Phelps, The Choquet representation in the complex case, Bull. Amer. Math. Soc. 83 (1977), no. 3, 299–312. MR 435818, DOI 10.1090/S0002-9904-1977-14243-2
Marc Rogalski, Caracterisation des simplexes par des propriétés portant sur les faces fermées et sur les ensembles compacts de points extrémaux, Math. Scand. 28 (1971), 159–181 (French). MR 306856, DOI 10.7146/math.scand.a-11013
Z. Semadeni, Free compact convex sets, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 141–146 (English, with Russian summary). MR 190719
Ulf Uttersrud, On $M$-ideals and the Alfsen-Effros structure topology, Math. Scand. 43 (1978), no. 2, 369–381 (1979). MR 531317, DOI 10.7146/math.scand.a-11792