𝑀-ideals, the strong 2-ball property and some renorming theorems

Yost, David

doi:10.1090/S0002-9939-1981-0593475-7

$M$-ideals, the strong $2$-ball property and some renorming theorems
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Proc. Amer. Math. Soc. 81 (1981), 299-303 Request permission

Abstract:

Examples are given of $M$-ideals in Banach spaces which do not possess the strong $2$-ball property. This solves a problem of Alfsen and Effros. A previous example is shown to be incorrect. The technique used to construct these examples is then employed to prove negative renorming theorems for Banach spaces. The following is representative: every separable Banach space has an equivalent norm which is strictly convex but not locally uniformly convex.

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
D. Amir and J. Lindenstrauss, The structure of weakly compact sets in Banach spaces, Ann. of Math. (2) 88 (1968), 35–46. MR 228983, DOI 10.2307/1970554
Dennis F. Cudia, The geometry of Banach spaces. Smoothness, Trans. Amer. Math. Soc. 110 (1964), 284–314. MR 163143, DOI 10.1090/S0002-9947-1964-0163143-9
M. Ĭ. Kadec′, On strong and weak convergence, Dokl. Akad. Nauk SSSR 122 (1958), 13–16 (Russian). MR 0098977
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
A. R. Lovaglia, Locally uniformly convex Banach spaces, Trans. Amer. Math. Soc. 78 (1955), 225–238. MR 66558, DOI 10.1090/S0002-9947-1955-0066558-7
R. I. Ovsepian and A. Pełczyński, On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space, and related constructions of uniformly bounded orthonormal systems in $L^{2}$, Studia Math. 54 (1975), no. 2, 149–159. MR 394137, DOI 10.4064/sm-54-2-149-159
Mark A. Smith, Some examples concerning rotundity in Banach spaces, Math. Ann. 233 (1978), no. 2, 155–161. MR 482080, DOI 10.1007/BF01421923
V. L. Šmulian, Sur la structure de la sphère unitaire dans l’espace de Banach, Rec. Math. [Mat. Sbornik] N.S. 9 (51) (1941), 545–561 (French). MR 0005775
S. L. Troyanski, On locally uniformly convex and differentiable norms in certain non-separable Banach spaces, Studia Math. 37 (1970/71), 173–180. MR 306873, DOI 10.4064/sm-37-2-173-180
David T. Yost, Best approximation and intersections of balls in Banach spaces, Bull. Austral. Math. Soc. 20 (1979), no. 2, 285–300. MR 557239, DOI 10.1017/S0004972700010972