On the Cauchy–Rassias stability of the trif functional equation in \(C^{*}\)-algebras. (English) Zbl 1064.46039
Let \(A\) and \(B\) be two unital \(C^*\)-algebras and let \(q:=k(n-1)/(n-k)\) for given integers \(k, n\) with \(2\leq k\leq n-1\). By an almost unital mapping we mean a mapping \(h:A\to B\) with the property that \({\lim_{j\to\infty}}q^jh(q^{-j}e)\) is invertible where \(e\) is the unit of \(A\).
The authors prove that every almost unital approximately linear mapping \(h:A\to B\) is a homomorphism if \(h(q^{-j}xu)=h(x)h(q^{-j}u)\) for all \(x\in A\), all unitaries \(u\in A\) and all sufficiently large integers \(j\). The proof is based on the fact that \(T(x):={\lim_{j\to\infty}}q^jh(q^{-j}x), x\in A\) exists and defines a homomorphism \(T\) and that \(T=h\).
T. Trif [J. Math. Anal. Appl. 272, No. 2, 604–616 (2002; Zbl 1036.39021)] investigated the Cauchy–Rassias stability of the so-called Trif functional equation. Using his result, the authors study the stability of Trif equations associated with \(*\)-homomorphisms between \(C^*\)-algebras and \(*\)-derivations of a unital \(C^*\)-algebra.
The authors prove that every almost unital approximately linear mapping \(h:A\to B\) is a homomorphism if \(h(q^{-j}xu)=h(x)h(q^{-j}u)\) for all \(x\in A\), all unitaries \(u\in A\) and all sufficiently large integers \(j\). The proof is based on the fact that \(T(x):={\lim_{j\to\infty}}q^jh(q^{-j}x), x\in A\) exists and defines a homomorphism \(T\) and that \(T=h\).
T. Trif [J. Math. Anal. Appl. 272, No. 2, 604–616 (2002; Zbl 1036.39021)] investigated the Cauchy–Rassias stability of the so-called Trif functional equation. Using his result, the authors study the stability of Trif equations associated with \(*\)-homomorphisms between \(C^*\)-algebras and \(*\)-derivations of a unital \(C^*\)-algebra.
Reviewer: Mohammad Sal Moslehian (Mashad-Delhi)
MSC:
| 46L05 | General theory of \(C^*\)-algebras |
| 39B82 | Stability, separation, extension, and related topics for functional equations |
| 39B52 | Functional equations for functions with more general domains and/or ranges |
Keywords:
Cauchy–Rassias stability; Trif functional equation; \(C^*\)-algebra; \(*\)-homomorphism; \(*\)-derivation; real rank zeroCitations:
Zbl 1036.39021References:
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