A pair (

$\mathcal{A},\mathcal{B})$ of Banach algebras is said to have the property AMNM (almost multiplicative maps are near multiplicative maps), if on bounded subsets of L(

$\mathcal{A},\mathcal{B})$ (the Banach space of bounded linear operators from

$\mathcal{A}$ into

$\mathcal{B})$ for any

$\u03f5>0$ there exists a

$\delta <0$ such that for any

$T\in L(\mathcal{A},\mathcal{B})$ the inequality

$\parallel T\left(ab\right)-T\left(a\right)T\left(b\right)\parallel \le \delta \parallel a\parallel \parallel b\parallel (a,b\in \mathcal{A})$ implies

$\parallel T-{T}^{\text{'}}\parallel \le \u03f5$ for some multiplicative map T’

$\in L(\mathcal{A},\mathcal{B})$. This paper is devoted to the question, which pairs of Banach algebras are AMNM pairs. As a central result this property is proven, when

$\mathcal{A}$ 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

$\mathcal{B}$ is the dual of a

$\mathcal{B}$-bimodule. This leads to results for the combination of group algebras with commutative algebras. Further positive answers are obtained for the case where

$\mathcal{B}$ is the algebra of all continuous functions on a compact Hausdorff space. Finally it is shown that the property AMNM holds, if

$\mathcal{A}$ and

$\mathcal{B}$ 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

${H}^{n}$ analogous to the Phragmén-Lindelöf algebra (this is due to the nontriviality of the center of the group

${H}^{n})$.