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

Olav Nygaard¹ ; Märt Põldvere²

¹ Department of Mathematics Agder University Servicebox 422 4604 Kristiansand, Norway
² Institute of Mathematics University of Tartu J. Liivi 2 50409 Tartu, Estonia

Studia Mathematica, Tome 195 (2009) no. 3, pp. 243-255

Résumé

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 $\mathcal{L}(X,X)$.

Détail

DOI : 10.4064/sm195-3-4

Keywords: banach spaces non separable proof kalton werner lima oja theorem subspace mathcal compact operators forms m ideal space mathcal continuous linear operators only has kaltons property ast metric compact approximation property proof quick consequence main results first describe johnsons projection mathcal ast applies mathcal ast represented via borel respect relative weak ast topology measure overline ast ast otimes ast ast subset mathcal ast ast has radon nikod property passes under integral sign basic theorem route description structure theorem borel probability measures overline ast ast otimes ast ast yields description mathcal ast due feder saphar second property ast equivalent every functional overline ast ast otimes ast ast behaving mathcal m ideal mathcal