A pair (${\cal A},{\cal B})$ of Banach algebras is said to have the property AMNM (almost multiplicative maps are near multiplicative maps), if on bounded subsets of L(${\cal A},{\cal B})$ (the Banach space of bounded linear operators from ${\cal A}$ into ${\cal B})$ for any $\epsilon >0$ there exists a $\delta <0$ such that for any $T\in L({\cal A},{\cal B})$ the inequality $\Vert T(ab)-T(a)T(b)\Vert \le \delta \Vert a\Vert \Vert b\Vert (a,b\in {\cal A})$ implies $\Vert T-T'\Vert \le \epsilon$ for some multiplicative map T’$\in L({\cal A},{\cal B})$. This paper is devoted to the question, which pairs of Banach algebras are AMNM pairs. As a central result this property is proven, when ${\cal A}$ is an amenable algebra (these are studied by the author in [Cohomology in Banach algebras, Mem. Am. Math. soc. 127 (1972;

Zbl 0256.18014)]) and ${\cal B}$ is the dual of a ${\cal B}$-bimodule. This leads to results for the combination of group algebras with commutative algebras. Further positive answers are obtained for the case where ${\cal B}$ is the algebra of all continuous functions on a compact Hausdorff space. Finally it is shown that the property AMNM holds, if ${\cal A}$ and ${\cal B}$ both equal to the algebra of all bounded linear operators on a separable Hilbert space. A corresponHeisenberg group. This class is substantially larger than in the one-dimensional case, but the additional condition of invariance under affine automorphisms distinguishes two nontrivial algebras on $H\sp n$ analogous to the Phragmén-Lindelöf algebra (this is due to the nontriviality of the center of the group $H\sp n)$.