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

• Autorzy: Olav Nygaard , Märt Põldvere
• Języki publikacji: EN

Abstrakty

EN

Let X and Y be Banach spaces. We give a "non-separable" proof of the Kalton-Werner-Lima-Oja theorem that the subspace 𝒦(X,X) of compact operators forms an M-ideal in the space 𝓛(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 𝓛(X,Y)* applies to f ∈ 𝓛(X,Y)* when f is represented via a Borel (with respect to the relative weak* topology) measure on $\overline{B_{X**} ⊗ B_{Y*}}^{w*} ⊂ 𝓛(X,Y)*$: 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 𝒦(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 𝒦(X,X) were an M-ideal in 𝓛(X,X).

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2009
• Tom: 195
• Numer: 3
• Strony: 243-255

Daty

wydano

2009

Twórcy

autor

Olav Nygaard

• Department of Mathematics, Agder University, Servicebox 422, 4604 Kristiansand, Norway

autor

Märt Põldvere

• Institute of Mathematics, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia

Identyfikatory

DOI

10.4064/sm195-3-4