Calkin representations for $L^p$

Abstract

We identify the weak closures of the ranges of certain Calkin representations for $L^p$, $1<p<\infty$. As a consequence, assuming the continuum hypothesis, we show that the commutant of $B(L^p)$, $1<p<\infty$, in its ultrapower may or may not be trivial depending on the ultrafilter. This extends a result of Farah, Phillips and Steprāns.