Approximately zero-product-preserving maps
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 \textit{B.\,E.\thinspace 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 [\textit{J.\,Alaminos, M.\,Brešar, J.~Extremera} and \textit{A.\,R.\thinspace 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.
approximately zero product preserving map
scientific article; zbMATH DE number 5823064
scientific article