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A fixed point approach to the stability of φ-morphisms on Hilbert C * -modules. (English) Zbl 1221.39034

Let E, F be Hilbert modules over C * -algebras A, B, respectively, and let φ:AB be a map. We say that U:EF is a φ-morphism if


Using a fixed point approach the authors prove stability results for φ-morphisms under the assumption of the form


where ρ(x,y) is a “control” function satisfying some technical conditions. In particular, for ρ(x,y)=cx p y p (with some c,p0 and p1), there exists a unique φ-morphism T:EF such that

U(x)-T(x)c(2 p +2) |2-2 p |x p forxE·

If p<0 both inequalites (in the assumption and in the assertion) are postulated for x,yE{0}.

39B82Stability, separation, extension, and related topics
46L08C * -modules
39B52Functional equations for functions with more general domains and/or ranges