Calkin representations for $L^{p}$

TL;DR

This paper investigates the structure of Calkin representations on L^p spaces, revealing how their weak closures behave and how the commutant in ultrapowers can vary under the continuum hypothesis.

Contribution

It identifies the weak closures of Calkin representation ranges on L^p spaces and shows the non-triviality of the commutant in ultrapowers depends on the ultrafilter, extending previous results.

Findings

Weak closures of Calkin representations on L^p are characterized.

The commutant in ultrapowers can be trivial or non-trivial depending on the ultrafilter.

Results depend on the continuum hypothesis.

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.