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Quantifying (non-)weak compactness of operators on
$AL$- and 
$C(K)$-spaces
- Part of
-
- Journal:
- Proceedings of the Edinburgh Mathematical Society , First View
- Published online by Cambridge University Press:
- 06 March 2026, pp. 1-56
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- Article
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We study the representation of non-weakly compact operators between
$AL$-spaces. In this setting, we show that every operator admits a best approximant in the ideal of weakly compact operators. Using duality arguments, we extend this result to operators between 
$C(L)$-spaces where 
$L$ is extremally disconnected. We also characterize the weak essential norm for operators between 
$AL$-spaces in terms of factorizations of the identity on 
$\ell_1$. As a consequence, we deduce that the weak Calkin algebra 
$\mathscr{B}(E)/\mathscr{W}(E)$ admits a unique algebra norm for every 
$AL$-space 
$E$. By duality, similar results are obtained for 
$C(K)$-spaces. In particular, we prove that for operators 
$T: L_{\infty}[0,1] \to L_{\infty}[0,1]$ the weak essential norm, the residuum norm, and the De Blasi measure of weak compactness coincide, answering a question of González, Saksman and Tylli.
