The Rademacher quotient of $L^1$ embeds into $L^1$ and its weak Calkin algebra is uniformly incompressible

Let $R$ be the closed linear span of the Rademacher functions in $L^1(0,1)$. We prove that $L^1(0,1)\mathrm{/}R$ embeds isomorphically into $L^1(0,1)$, answering affirmatively a question of González and Martínez-Abejón \cite[p.~140]{GM97}. We then use this embedding to show that the weak Calkin algebra of the quotient is uniformly incompressible. \end{abstract