## Approximately multiplicative maps between Banach algebras.(English)Zbl 0652.46031

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 $$\epsilon >0$$ there exists a $$\delta <0$$ such that for any $$T\in L({\mathcal A},{\mathcal B})$$ the inequality $$\| T(ab)-T(a)T(b)\| \leq \delta \| a\| \| b\| (a,b\in {\mathcal A})$$ implies $$\| T-T'\| \leq \epsilon$$ 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 $$H^ n)$$.
Reviewer: J.B.Prolla

### MSC:

 46H05 General theory of topological algebras 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)

Zbl 0256.18014
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