Automorphisms on a \(C^*\)-algebra and isomorphisms between Lie \(JC^*\)-algebras associated with a generalized additive mapping. (English) Zbl 1135.39016
The author considers the following functional equation
\[ \begin{split} rf\bigg(\frac{\sum_{j=1}^d x_j}{r}\biggr)+ \sum_{s(j)=0,1; \sum_{j=1}^d s(j)=l} rf \biggl(\frac{\sum_{j=1}^d (-1)^{s(j)}x_j}{r}\biggr)=\\ \biggl[\binom{l}{d-1}-\binom{l-1}d -1+1\biggr] \sum_{j=1}^d f(x_j),\end{split}\tag{*} \]
where \(f:X \to Y\) (\(X\), \(Y\) vector spaces) is an odd mapping, \(r\) is \(1\) or \(3\), and \(d\) and \(l\) are integers with \(1<l<d/2\). After proving that the solutions of equation (*) are the additive functions, some results of stability, in the sense of Ulam-Hyers, are proved. These are then used for proving that some odd bijective mappings from a unital \(C^{*}\)-algebra A into itself satisfying certain inequalities are automorphisms.
\[ \begin{split} rf\bigg(\frac{\sum_{j=1}^d x_j}{r}\biggr)+ \sum_{s(j)=0,1; \sum_{j=1}^d s(j)=l} rf \biggl(\frac{\sum_{j=1}^d (-1)^{s(j)}x_j}{r}\biggr)=\\ \biggl[\binom{l}{d-1}-\binom{l-1}d -1+1\biggr] \sum_{j=1}^d f(x_j),\end{split}\tag{*} \]
where \(f:X \to Y\) (\(X\), \(Y\) vector spaces) is an odd mapping, \(r\) is \(1\) or \(3\), and \(d\) and \(l\) are integers with \(1<l<d/2\). After proving that the solutions of equation (*) are the additive functions, some results of stability, in the sense of Ulam-Hyers, are proved. These are then used for proving that some odd bijective mappings from a unital \(C^{*}\)-algebra A into itself satisfying certain inequalities are automorphisms.
Reviewer: Gian Luigi Forti (Milano)
MSC:
39B82 | Stability, separation, extension, and related topics for functional equations |
39B52 | Functional equations for functions with more general domains and/or ranges |
46L05 | General theory of \(C^*\)-algebras |
47B48 | Linear operators on Banach algebras |
17B40 | Automorphisms, derivations, other operators for Lie algebras and super algebras |