Johnson's projection, Kalton's property $(M^\ast)$, and $M$-ideals of compact operators
Tom 195 / 2009
Streszczenie
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 $\mathcal{L}(X,X)$.
