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 Stepr\=ans.