Every nonreflexive subspace of 𝐿₁[0,1] fails the fixed point property

Dowling, P.; Lennard, C.

doi:10.1090/S0002-9939-97-03577-6

Every nonreflexive subspace of $L_1[0,1]$ fails the fixed point property
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by P. N. Dowling and C. J. Lennard PDF

Proc. Amer. Math. Soc. 125 (1997), 443-446 Request permission

Abstract:

The main result of this paper is that every nonreflexive subspace $Y$ of $L_{1}[0,1]$ fails the fixed point property for closed, bounded, convex subsets $C$ of $Y$ and nonexpansive (or contractive) mappings on $C$. Combined with a theorem of Maurey we get that for subspaces $Y$ of $L_{1}[0,1]$, $Y$ is reflexive if and only if $Y$ has the fixed point property. For general Banach spaces the question as to whether reflexivity implies the fixed point property and the converse question are both still open.

References

Dale E. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82 (1981), no. 3, 423–424. MR 612733, DOI 10.1090/S0002-9939-1981-0612733-0
N.L. Carothers, S.J. Dilworth and C.J. Lennard, On a localization of the UKK property and the fixed point property in $L_{w,1}$, Interaction Between Functional Analysis, Harmonic Analysis, and Probability (Columbia, MO, 1994), Lecture Notes in Pure and Appl. Math., vol. 175, Dekker, New York, 1966, pp. 111–124.
Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
P.G. Dodds, T.K. Dodds, P.N. Dowling and C.J. Lennard, Asymptotically isometric copies of $\ell _{1}$ in nonreflexive subspaces of symmetric operator spaces, in preparation.
P.N. Dowling, W.B. Johnson, C.J. Lennard and B. Turett, The optimality of James’s distortion theorems, Proc. Amer. Math. Soc. (to appear).
P.N. Dowling, C.J. Lennard and B. Turett, Reflexivity and the fixed point property for nonexpansive maps, J. Math. Anal. Appl. 200 (1996), 653–652.
J. Elton, Pei-Kee Lin, E. Odell, and S. Szarek, Remarks on the fixed point problem for nonexpansive maps, Fixed points and nonexpansive mappings (Cincinnati, Ohio, 1982) Contemp. Math., vol. 18, Amer. Math. Soc., Providence, RI, 1983, pp. 87–120. MR 728595, DOI 10.1090/conm/018/728595
M. I. Kadec and A. Pełczyński, Bases, lacunary sequences and complemented subspaces in the spaces $L_{p}$, Studia Math. 21 (1961/62), 161–176. MR 152879, DOI 10.4064/sm-21-2-161-176
Teck Cheong Lim, Asymptotic centers and nonexpansive mappings in conjugate Banach spaces, Pacific J. Math. 90 (1980), no. 1, 135–143. MR 599326
Kichi-Suke Saito, Corrigendum: “Invariant subspaces and cocycles in nonselfadjoint crossed products” [J. Funct. Anal. 45 (1982), no. 2, 177–193; MR0647070 (84a:46141)], J. Funct. Anal. 61 (1985), no. 1, 116. MR 779741, DOI 10.1016/0022-1236(85)90041-2
B. Maurey, Points fixes des contractions de certains faiblement compacts de $L^{1}$, Seminar on Functional Analysis, 1980–1981, École Polytech., Palaiseau, 1981, pp. Exp. No. VIII, 19 (French). MR 659309
M. Smyth, Remarks on the weak star fixed point property in the dual of $C(\Omega )$, J. Math. Anal. Appl. 195 (1995), 294–306.