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Approximately multiplicative maps between Banach algebras. (English) Zbl 0652.46031
A pair (𝒜,) of Banach algebras is said to have the property AMNM (almost multiplicative maps are near multiplicative maps), if on bounded subsets of L(𝒜,) (the Banach space of bounded linear operators from 𝒜 into ) for any ϵ>0 there exists a δ<0 such that for any TL(𝒜,) the inequality T(ab)-T(a)T(b)δab(a,b𝒜) implies T-T ' ϵ for some multiplicative map T’L(𝒜,). 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 is the dual of a -bimodule. This leads to results for the combination of group algebras with commutative algebras. Further positive answers are obtained for the case where is the algebra of all continuous functions on a compact Hausdorff space. Finally it is shown that the property AMNM holds, if 𝒜 and 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:
46H05General theory of topological algebras
46H25Normed modules and Banach modules, topological modules