Stability of multipliers on Banach algebras (Q1777702)

Let \(A\) be a Banach algebra. A mapping \(T:A\to A\) is called a multiplier if \(aT(b)=T(a)b\) for all \(a,b\in A\). Recall that a Banach algebra \(A\) is without order if for \(x\in A\), \(xA=\{0\}\) implies \(x=0\), or, \(Ax=\{0\}\) implies \(x=0\). The authors show that if \(T\) is a mapping with \(T(0)=0\) on a complex Banach algebra without order, \(\| aT(b)-T(a)b\| \leq \epsilon\| a\| ^p\| b\| ^p\) for some \(\epsilon>0\), \(p\neq 1\), and all \(a,b\in A\), then \(T\) is a multiplier (for the case \(p>0\), we do not need the assumption \(T(0)=0\)). Giving an example, they show that a similar result need not be true for \(p=1\). In addition, they show that if \(f\) is an approximate multiplier which is also approximately additive, then there is a multiplier near to \(f\), and thus prove the Hyers--Ulam--Rassias stability of multipliers. For an extensive account of Hyers--Ulam--Rassias stability of various functional equations, one may refer to \textit{D. H. Hyers, G. Isac} and \textit{Th. M. Rassias} [``Stability of functional equations in several variables'' (Progress in Nonlinear Differential Equations and their Applications 34, Birkhäuser Boston, MA) (1998; Zbl 0907.39025)].