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Stability of linear mappings in quasi-Banach modules. (English) Zbl 1154.39028
Let ${\mathcal A}$ be a unital $C^\ast$-algebra and ${\mathcal A}_1^+$ be the positive part of the unit ball ${\mathcal A}_1$ of ${\mathcal A}$. Let ${\mathcal X}$ be a quasi-normed ${\mathcal A}$-module with quasi-norm $\Vert \cdot\Vert _{\mathcal X}$ and ${\mathcal Y}$ be a $p$-Banach ${\mathcal A}$-module with $p$-norm $\Vert \cdot\Vert _{\mathcal Y}$ (i.e. $0<p\leq 1$ and $\Vert x+y\Vert _{\mathcal Y}^p\leq\Vert x\Vert _{\mathcal Y}^p+\Vert y\Vert _{\mathcal Y}^p$ for $x,y\in {\mathcal Y}$). The authors prove the Hyers-Ulam-Rassias stability of linear mappings from ${\mathcal X}$ to ${\mathcal Y}$, associated to the Cauchy functional equation and a generalized version of the Jensen equation. In the first case they assume that a function $f:{\mathcal X}\to {\mathcal Y}$ fulfils $$\Vert f(ux+y)-uf(x)-f(y)\Vert _{\mathcal Y}\leq\varepsilon (\Vert x\Vert _{\mathcal X}^r+\Vert y\Vert _{\mathcal X}^r)$$ for $x,y\in {\mathcal X}$ and for all $u$ from the unitary group $U({\mathcal A})$ if $r>1$, and for all $u\in {\mathcal A}_1^+\cup\{\text{ i}\}$ if $r<1$. In the second case they assume that $f(0)=0$ and for some integer $N>1$ we have $$\left\Vert Nf\left({ax+y\over N}\right)-af(x)-f(y)\right\Vert _{\mathcal Y}\leq\varepsilon (\Vert x\Vert _{\mathcal X}^r+\Vert y\Vert _{\mathcal X}^r)$$ for $x,y\in {\mathcal X}$ and for all $a$ from the unit sphere of ${\mathcal A}$. In both cases the authors show that there is a unique ${\mathcal A}$-linear mapping $L:{\mathcal X}\to {\mathcal Y}$ satisfying $$\Vert f(x)-L(x)\Vert _{\mathcal Y}\leq K(\varepsilon ,p, r, N)\Vert x\Vert _{\mathcal X}^r\quad (x\in {\mathcal X})$$ with certain stability constants $K(\varepsilon ,p, r, N)$ which always tend to zero as $\varepsilon\to 0$.

39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
46L05General theory of $C^*$-algebras