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Approximately zero-product-preserving maps. (English) Zbl 1209.47013

A continuous linear map \(T\) from a Banach algebra \(A\) into \(B\) approximately preserves the zero products if \(\|T(a)T(b)\| \leq \alpha \|a\| \|b\|\) for all \(a, b \in A\) with \(ab= 0\), with some small positive \(\alpha\). In the context of function algebras, it is known that a zero-product preserving map, which is also known as a disjointness preserving map or separation map, is a weighted homomorphism \(W\Phi\), where \(\Phi: A \to B\) is a homomorphism and \(W: B \to B\) is a \(B\)-bimodule homomorphism.
This paper is mainly concerned with the question of whether a continuous linear surjective approximately zero-product preserving map is close to a continuous weighted homomorphism. To attack this question, for such a map \(T\), the authors associate with \(T\) a continuous bilinear map \(\phi: A \times A \to B\) defined by \(\phi(a, b):= T(a) T(b)\) which obviously satisfies the condition
\[ a, b \in A, \quad ab= 0 \Longrightarrow \|\phi(a, b)\| \leq \alpha \|a\| \|b\|. \]
They show that in the case where \(A\) is a group algebra or a \(C^*\)-algebra and the multiplier algebra of \(B\) is a dual space, any continuous linear surjective approximately zero-product preserving map is of the form \(WS\) for some invertible \(B\)-bimodule homomorphism \(W: B \to B\) and some continuous linear approximately multiplicative surjection \(S: A \to B\).
Approximately multiplicative maps were studied by B.E.Johnson [J. Lond.Math.Soc., II.Ser., 37, No.2, 294–316 (1988; Zbl 0652.46031)] who showed that whenever \(A\) is amenable, an approximately multiplicative map is close to a multiplicative map. On the other hand, the continuous bilinear maps \(\phi\) satisfying the equation with \(\alpha= 0\) have been studied in [J.Alaminos, M.Brešar, J. Extremera and A.R.Villena, Proc.R.Soc.Edinb., Sect.A, 137, No.1, 1–7 (2007; Zbl 1144.47030) and Stud.Math.193, No.2, 131–159 (2009; Zbl 1168.47029)] as a powerful tool for characterizing both homomorphisms and derivations through the zero-product analysis.

MSC:

47B48 Linear operators on Banach algebras
47A07 Forms (bilinear, sesquilinear, multilinear)
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References:

[1] J. Alaminos, M. Brešar, J. Extremera and A. R. Villena, Characterizing homomorphisms and derivations on C*-algebras, Proceedings of the Royal Society of Edinburgh. Section A 137 (2007), 1–7. · Zbl 1144.47030
[2] J. Alaminos, M. Brešar, J. Extremera and A. R. Villena, Maps preserving zero products, Studia Mathematica 193 (2009), 131–159. · Zbl 1168.47029 · doi:10.4064/sm193-2-3
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