## Facial characterizations of complex Lindenstrauss spaces

HTML articles powered by AMS MathViewer

- 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

## Additional Information

- © Copyright 1981 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**268**(1981), 173-186 - MSC: Primary 46B10; Secondary 46A55
- DOI: https://doi.org/10.1090/S0002-9947-1981-0628453-7
- MathSciNet review: 628453