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

Olav Nygaard, and Märt Põldvere. "Johnson's projection, Kalton's property (M*), and M-ideals of compact operators." Studia Mathematica 195.3 (2009): 243-255. <http://eudml.org/doc/284738>.

@article{OlavNygaard2009,
abstract = {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).},
author = {Olav Nygaard, Märt Põldvere},
journal = {Studia Mathematica},
keywords = {Johnson's projection; Kalton’s property ; dual of compact operators; M-ideals of compact operator},
language = {eng},
number = {3},
pages = {243-255},
title = {Johnson's projection, Kalton's property (M*), and M-ideals of compact operators},
url = {http://eudml.org/doc/284738},
volume = {195},
year = {2009},
}

TY - JOUR
AU - Olav Nygaard
AU - Märt Põldvere
TI - Johnson's projection, Kalton's property (M*), and M-ideals of compact operators
JO - Studia Mathematica
PY - 2009
VL - 195
IS - 3
SP - 243
EP - 255
AB - 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).
LA - eng
KW - Johnson's projection; Kalton’s property ; dual of compact operators; M-ideals of compact operator
UR - http://eudml.org/doc/284738
ER -