  We show that for any separable reflexive Banach space $X$ and a large class
of Banach spaces $E$, including those with a subsymmetric shrinking basis but
also all spaces $L_p$ for $1\leq p \leq \infty$, every bounded linear map
${\mathcal B}(E)\to {\mathcal B}(X)$ which is approximately multiplicative is
necessarily close in the operator norm to some bounded homomorphism ${\mathcal
B}(E)\to {\mathcal B}(X)$. That is, the pair $({\mathcal B}(E), {\mathcal
B}(X))$ has the AMNM property in the sense of Johnson (\textit{J.~London Math.\
Soc.} 1988). Previously this was only known for $E=X=\ell_p$ with $1<p<\infty$;
even for those cases, we improve on the previous methods and obtain better
constants in various estimates. A crucial role in our approach is played by a
new result, motivated by cohomological techniques, which establishes AMNM
properties relative to an amenable subalgebra; this generalizes a theorem of
Johnson (\textit{op cit.}).
