Johnson's projection, Kalton's property (M*), and M-ideals of compact operators

Olav Nygaard ; Märt Põldvere

Studia Mathematica, Tome 192 (2009), p. 243-255 / Harvested from The Polish Digital Mathematics Library

Résumé

Let X and Y be Banach spaces. We give a “non-separable” proof of the Kalton-Werner-Lima-Oja theorem that the subspace (X,X) of compact operators forms an M-ideal in the space (X,X) of all continuous linear operators from X to X if and only if X has Kalton’s property (M*) 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 (X,Y)* applies to f ∈ (X,Y)* when f is represented via a Borel (with respect to the relative weak* topology) measure on $\overline{B_{X**} ⊗ B_{Y*}}^{w*} ⊂ (X,Y)*$: If Y* 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 $\overline{B_{X**} ⊗ B_{Y*}}^{w*}$-also yields a description of (X,Y)* due to Feder and Saphar. Second, we show that property (M*) for X is equivalent to every functional in $\overline{B_{X**} ⊗ B_{X*}}^{w*}$ behaving as if (X,X) were an M-ideal in (X,X).

Détail

Publié le : 2009-01-01

Zbl 1192.46011

EUDML-ID : urn:eudml:doc:284738