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Approximately multiplicative maps between Banach algebras. (English) Zbl 0652.46031
A pair ($𝒜,ℬ\right)$ of Banach algebras is said to have the property AMNM (almost multiplicative maps are near multiplicative maps), if on bounded subsets of L($𝒜,ℬ\right)$ (the Banach space of bounded linear operators from $𝒜$ into $ℬ\right)$ for any $ϵ>0$ there exists a $\delta <0$ such that for any $T\in L\left(𝒜,ℬ\right)$ the inequality $\parallel T\left(ab\right)-T\left(a\right)T\left(b\right)\parallel \le \delta \parallel a\parallel \parallel b\parallel \left(a,b\in 𝒜\right)$ implies $\parallel T-{T}^{\text{'}}\parallel \le ϵ$ for some multiplicative map T’$\in L\left(𝒜,ℬ\right)$. This paper is devoted to the question, which pairs of Banach algebras are AMNM pairs. As a central result this property is proven, when $𝒜$ 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 $ℬ$ is the dual of a $ℬ$-bimodule. This leads to results for the combination of group algebras with commutative algebras. Further positive answers are obtained for the case where $ℬ$ is the algebra of all continuous functions on a compact Hausdorff space. Finally it is shown that the property AMNM holds, if $𝒜$ and $ℬ$ 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 ${H}^{n}\right)$.
Reviewer: J.B.Prolla

##### MSC:
 46H05 General theory of topological algebras 46H25 Normed modules and Banach modules, topological modules