We show that for any separable reflexive Banach space
X
X
and a large class of Banach spaces
E
E
, including those with a subsymmetric shrinking basis but also all spaces
L
p
[
0
,
1
]
L_p[0,1]
for
1
≤
p
≤
∞
1\le p \le \infty
, every bounded linear map
B
(
E
)
→
B
(
X
)
\mathcal {B}(E)\to \mathcal {B}(X)
which is approximately multiplicative is necessarily close in the operator norm to some bounded homomorphism
B
(
E
)
→
B
(
X
)
\mathcal {B}(E)\to \mathcal {B}(X)
. That is, the pair
(
B
(
E
)
,
B
(
X
)
)
(\mathcal {B}(E),\mathcal {B}(X))
has the AMNM property in the sense of Johnson [J. London Math. Soc. (2) 37 (1988), pp. 294–316]. Previously this was only known for
E
=
X
=
ℓ
p
E=X=\ell _p
with
1
>
p
>
∞
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 (op cit.).