Calkin representations for $L^p$

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