Centralizers on a super-reflexive Schatten ideal
Jesús Suárez
TL;DR
The paper settles the question of whether strictly singular bicentralizers can exist on super-reflexive Schatten ideals, proving they cannot, and in particular for the $p$-Schatten class $C_{\ell_p}$ with $1<p<\infty$. It relies on Kalton's insight that centralizers on Schatten classes arise from interpolation, and reconstructs a Calderón interpolation between symmetric Köthe sequence spaces to extract a copy of $\ell_2$ inside the ideal. By constraining the bicentralizer to act boundedly on this $\ell_2$-copy via its Hermitian components, the paper shows any bicentralizer cannot be strictly singular. The result extends known $L_p$-space phenomena to Schatten ideals and provides a streamlined argument based on interpolation and operator-space structure.
Abstract
We give a simple proof that there is no strictly singular bicentralizer on a super-reflexive Schatten ideal. This result applies, in particular, to the $p$-Schatten class for $1<p<\infty$.