A+ CATEGORY SCIENTIFIC UNIT

Johnson's projection, Kalton's property $(M^\ast)$, and $M$-ideals of compact operators

Volume 195 / 2009

Olav Nygaard, Märt Põldvere Studia Mathematica 195 (2009), 243-255 MSC: Primary 46B20; Secondary 46B28. DOI: 10.4064/sm195-3-4

Abstract

Let $X$ and $Y$ be Banach spaces. We give a “non-separable” proof of the Kalton–Werner–Lima–Oja theorem that the subspace $\mathcal{K}(X,X)$ of compact operators forms an $M$-ideal in the space $\mathcal{L}(X,X)$ of all continuous linear operators from $X$ to $X$ if and only if $X$ has Kalton's property $(M^\ast)$ and the metric compact approximation property. Our proof is a quick consequence of two main results. First, we describe how Johnson's projection $P$ on $\mathcal{L}(X,Y)^\ast$ applies to $f\in\mathcal{L}(X,Y)^\ast$ when $f$ is represented via a Borel (with respect to the relative weak$^\ast$ topology) measure on $\overline{B_{{X^{\ast\ast}}}\otimes B_{Y^{\ast}}}^{w^\ast}\subset\mathcal{L}(X,Y)^\ast$: If $Y^{\ast}$ has the Radon–Nikodým property, then $P$ “passes under the integral sign”. Our basic theorem en route to this description—a structure theorem for Borel probability measures on $\overline{B_{{X^{\ast\ast}}}\otimes B_{Y^{\ast}}}^{w^\ast}\!$—also yields a description of $\mathcal{K}(X,Y)^\ast$ due to Feder and Saphar. Second, we show that property $(M^\ast)$ for $X$ is equivalent to every functional in $\overline{B_{{X^{\ast\ast}}}\otimes B_{X^{\ast}}}^{w^\ast}$ behaving as if $\mathcal{K}(X,X)$ were an $M$-ideal in $\mathcal{L}(X,X)$.

Authors

  • Olav NygaardDepartment of Mathematics
    Agder University
    Servicebox 422
    4604 Kristiansand, Norway
    e-mail
  • Märt PõldvereInstitute of Mathematics
    University of Tartu
    J. Liivi 2
    50409 Tartu, Estonia
    e-mail

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