Infinite dimensional 𝐿-spaces do not have preduals of all orders

Armstrong, Thomas E.

doi:10.1090/S0002-9939-1979-0524301-0

Infinite dimensional $L$-spaces do not have preduals of all orders
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by Thomas E. Armstrong

Proc. Amer. Math. Soc. 74 (1979), 285-290

DOI: https://doi.org/10.1090/S0002-9939-1979-0524301-0

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Abstract:

It is shown that if E is an infinite dimensional Banach space with first dual E’, second dual E", and nth dual ${E^{[n]}}$ and if ${E^{[n]}}$ is either an L- or M-space all duals are either L- or M-spaces except possibly E which could be a Lindenstrauss space. If E is an L- or M-space there is an integer $n(E)$ so that if $m > n(E)$ there is no Banach space F with $E = {F^{[m]}}$. The linear isomorphic analogues to these isometric results are also established. In particular if E is an ${\mathcal {L}_1}$ or ${\mathcal {L}_\infty }$ space there is an integer $\bar n(E)$ so that E is not linearly isomorphic to ${F^{[m]}}$ for any Banach space F when $m > \bar n(E)$.

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, DOI 10.1007/978-3-642-65009-3
Michèle Capon, Étude des espaces $A(K)$ qui sont des duaux, Math. Scand. 32 (1973), 225–241 (1974) (French). MR 338728, DOI 10.7146/math.scand.a-11457
A. J. Ellis, The duality of partially ordered normed linear spaces, J. London Math. Soc. 39 (1964), 730–744. MR 169020, DOI 10.1112/jlms/s1-39.1.730

Abstrakte Funktionen und lineare Operatoren

46

On dual

spaces and injective bidual Banach spaces

Robert C. James, A non-reflexive Banach space isometric with its second conjugate space, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 174–177. MR 44024, DOI 10.1073/pnas.37.3.174
Joram Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964), 112. MR 179580
Joram Lindenstrauss, A remark on ${\cal L}_{1}$ spaces, Israel J. Math. 8 (1970), 80–82. MR 259582, DOI 10.1007/BF02771554
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR 0415253
A. Pełczyński, On Banach spaces containing $L_{1}(\mu )$, Studia Math. 30 (1968), 231–246. MR 232195, DOI 10.4064/sm-30-2-231-246
A. Pełczyński, On the isomorphism of the spaces $m$ and $M$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 6 (1958), 695–696. MR 0102727
Haskell P. Rosenthal, On complemented and quasi-complemented subspaces of quotients of $C(S)$ for Stonian $S$, Proc. Nat. Acad. Sci. U.S.A. 60 (1968), 1165–1169. MR 231185, DOI 10.1073/pnas.60.4.1165
Haskell P. Rosenthal, On injective Banach spaces and the spaces $L^{\infty }(\mu )$ for finite measure $\mu$, Acta Math. 124 (1970), 205–248. MR 257721, DOI 10.1007/BF02394572
Zbigniew Semadeni, Banach spaces of continuous functions. Vol. I, Monografie Matematyczne, Tom 55, PWN—Polish Scientific Publishers, Warsaw, 1971. MR 0296671