The Maurey extension property for Banach spaces with the Gordon–Lewis property and related structures

Volume 155 / 2003

P. G. Casazza, N. J. Nielsen Studia Mathematica 155 (2003), 1-21 MSC: 46B03, 46B07. DOI: 10.4064/sm155-1-1


The main result of this paper states that if a Banach space $X$ has the property that every bounded operator from an arbitrary subspace of $X$ into an arbitrary Banach space of cotype 2 extends to a bounded operator on $X$, then every operator from $X$ to an $L_1$-space factors through a Hilbert space, or equivalently $B(\ell _{\infty },X^*)={\mit \Pi }_2(\ell _{\infty },X^*)$. If in addition $X$ has the Gaussian average property, then it is of type 2. This implies that the same conclusion holds if $X$ has the Gordon–Lewis property (in particular $X$ could be a Banach lattice) or if $X$ is isomorphic to a subspace of a Banach lattice of finite cotype, thus solving the Maurey extension problem for these classes of spaces. The paper also contains a detailed study of the property of extending operators with values in $\ell _p$-spaces, $1\le p<\infty $.


  • P. G. CasazzaDepartment of Mathematics
    University of Missouri
    Columbia, MO 65211, U.S.A.
  • N. J. NielsenDepartment of Mathematics and Computer Science
    University of Southern Denmark
    Campusvej 55
    DK-5230 Odense M, Denmark

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