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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)

Citations:

Zbl 0256.18014
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