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Stability of isometries between groups of invertible elements in Banach algebras. (English. Russian original) Zbl 1332.46046

Funct. Anal. Appl. 49, No. 2, 106-109 (2015); translation from Funkts. Anal. Prilozh. 49, No. 2, 34-38 (2015).
Let \(X\) and \(Y\) be Banach spaces and \(\varepsilon>0\). A map \(f : X \to Y\) is called an \(\varepsilon\)-isometry if \(|\, \|f(x)-f(y)\| - \|x-y\|\,|\leq \varepsilon\) for all \(x, y \in X\). The authors prove the Hyers-Ulam stability of surjective isometries between groups of invertible elements of certain Banach algebras. More precisely, let \(A\) be a unital Banach algebra, and let \(B = C(K)\), where \(K\) is a compact metric space. Assume that \(\mathfrak{A}\) and \(\mathfrak{B}\) are open multiplicative subgroups of the groups of invertible elements \(A^{-1}\) and \(B^{-1}\), respectively. If \(f : \{0\}\cup \mathfrak{A} \to \{0\}\cup \mathfrak{B}\) is a surjective \(\varepsilon\)-isometry with \(f(0)=0\), then there is a surjective positive homogeneous isometry \(U :cl(\mathfrak{A}) \to cl(\mathfrak{B})\) between the closures such that \(\|f(a)-Ua\| \leq 4\varepsilon\) for all \(a\in \{0\}\cup \mathfrak{A}\).

MSC:

46H05 General theory of topological algebras
46B04 Isometric theory of Banach spaces
39B82 Stability, separation, extension, and related topics for functional equations
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