Real structure in complex 𝐿₁-preduals

Wulbert, Daniel E.

doi:10.1090/S0002-9947-1978-0472918-8

Real structure in complex $L_{1}$-preduals
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Trans. Amer. Math. Soc. 235 (1978), 165-181 Request permission

Abstract:

Call a complex Banach space selfadjoint if it is isometrically isomorphic to a selfadjoint subspace of a $C(X,{\mathbf {C}})$-space. B. Hirsberg and A. Lazar proved that if the unit ball of a complex Lindenstrauss space, E, has an extreme point, then E is selfadjoint. Here we will give a characterization of selfadjoint Lindenstrauss spaces, and construct a nonselfadjoint complex Lindenstrauss space.

References

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
Y. Benyamini and J. Lindenstrauss, A predual of $l_{1}$ which is not isomorphic to a $C(K)$ space, Israel J. Math. 13 (1972), 246–254 (1973). MR 331013, DOI 10.1007/BF02762798
Errett Bishop, A generalization of the Stone-Weierstrass theorem, Pacific J. Math. 11 (1961), 777–783. MR 133676
Jean Dieudonné, Sur les espaces $L^{1}$, Arch. Math. 10 (1959), 151–152 (French). MR 105016, DOI 10.1007/BF01240778
James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
Edward G. Effros, Structure in simplexes, Acta Math. 117 (1967), 103–121. MR 203435, DOI 10.1007/BF02395042
Edward G. Effros, On a class of complex Banach spaces, Illinois J. Math. 18 (1974), 48–59. MR 328548
Hicham Fakhoury, Préduaux de $L$-espace: Notion de centre, J. Functional Analysis 9 (1972), 189–207 (French). MR 0303260, DOI 10.1016/0022-1236(72)90010-9
A. Grothendieck, Une caractérisation vectorielle-métrique des espaces $L^1$, Canadian J. Math. 7 (1955), 552–561 (French). MR 76301, DOI 10.4153/CJM-1955-060-6
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
Shizuo Kakutani, Concrete representation of abstract $(M)$-spaces. (A characterization of the space of continuous functions.), Ann. of Math. (2) 42 (1941), 994–1024. MR 5778, DOI 10.2307/1968778
Shizuo Kakutani, Concrete representation of abstract $(L)$-spaces and the mean ergodic theorem, Ann. of Math. (2) 42 (1941), 523–537. MR 4095, DOI 10.2307/1968915
H. Elton Lacey, The isometric theory of classical Banach spaces, Die Grundlehren der mathematischen Wissenschaften, Band 208, Springer-Verlag, New York-Heidelberg, 1974. MR 0493279
A. J. Lazar and J. Lindenstrauss, On Banach spaces whose duals are $L_{1}$ spaces, Israel J. Math. 4 (1966), 205–207. MR 206670, DOI 10.1007/BF02760079
A. J. Lazar, The unit ball in conjugate $L_{1}$ spaces, Duke Math. J. 39 (1972), 1–8. MR 303242
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
D. R. Lewis and C. Stegall, Banach spaces whose duals are isomorphic to $l_{1}(\Gamma )$, J. Functional Analysis 12 (1973), 177–187. MR 0342987, DOI 10.1016/0022-1236(73)90022-0
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR 0415253
Joram Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964), 112. MR 179580
Joram Lindenstrauss and Daniel E. Wulbert, On the classification of the Banach spaces whose duals are $L_{1}$ spaces, J. Functional Analysis 4 (1969), 332–349. MR 0250033, DOI 10.1016/0022-1236(69)90003-2
Asvald Lima, Complex Banach spaces whose duals are $L_{1}$-spaces, Israel J. Math. 24 (1976), no. 1, 59–72. MR 425584, DOI 10.1007/BF02761429
E. Michael and A. Pełczyński, Peaked partition subspaces of $C(X)$, Illinois J. Math. 11 (1967), 555–562. MR 217581
E. Michael and A. Pełczyński, Separable Banach spaces which admit $l_{n}{}^{\infty }$ approximations, Israel J. Math. 4 (1966), 189–198. MR 211247, DOI 10.1007/BF02760077

Complex preduals of

and subspaces of

Gunner Hans Olsen, On the classification of complex Lindenstrauss-spaces, Mathematics Preprint Series, No. 16 (1973), Universitetet i Oslo, Matematisk Institutt, Oslo, 1973. MR 0394128
Robert R. Phelps, Lectures on Choquet’s theorem, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0193470
Christian Samuel, Sur certains espaces $C_{\sigma }(S)$ et sur les sous-espaces complémentés de $C(S)$, Bull. Sci. Math. (2) 95 (1971), 65–82 (French). MR 285898
C. Stegall, Banach spaces whose duals contain $l_{1}(\Gamma )$ with applications to the study of dual $L_{1}(\mu )$ spaces, Trans. Amer. Math. Soc. 176 (1973), 463–477. MR 315404, DOI 10.1090/S0002-9947-1973-0315404-3
Daniel E. Wulbert, Convergence of operators and Korovkin’s theorem, J. Approximation Theory 1 (1968), 381–390. MR 235370, DOI 10.1016/0021-9045(68)90016-6
M. Zippin, On some subspaces of Banach spaces whose duals are $L_{1}$ spaces, Proc. Amer. Math. Soc. 23 (1969), 378–385. MR 246094, DOI 10.1090/S0002-9939-1969-0246094-0