  Non-amenability of ${\mathcal B}(E)$ has been surprisingly difficult to prove
for the classical Banach spaces, but is now known for $E= \ell_p$ and $E=L_p$
for all $1\leq p<\infty$. However, the arguments are rather indirect: the proof
for $L_1$ goes via non-amenability of $\ell^\infty({\mathcal K}(\ell_1))$ and a
transference principle developed by Daws and Runde (Studia Math., 2010). In
this note, we provide a short proof that ${\mathcal B}(L_1)$ and some of its
subalgebras are non-amenable, which completely bypasses all of this machinery.
Our approach is based on classical properties of the ideal of representable
operators on $L_1$, and shows that ${\mathcal B}(L_1)$ is not even
approximately amenable.
