• Journal of the Chungcheong Mathematical Society
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• v.24 no.2
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• pp.221-238
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• 2011

We prove the Hyers-Ulam stability of generalized Jensen's equations in Banach modules over a unital $C^{\ast}$-algebra. It is applied to show the stability of generalized Jensen's equations in a Hilbert module over a unital $C^{\ast}$-algebra. Moreover, we prove the stability of linear operators in a Hilbert module over a unital $C^{\ast}$-algebra.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.2
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• pp.287-301
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• 2011

Let X and Y be vector spaces. It is shown that a mapping $f\;:\;X{\rightarrow}Y$ satisfies the functional equation $$(\ddag)\hspace{50}dk\;f\left(\frac{\sum_{j=1}^{dk}x_j}{dk}\right)=\displaystyle\sum_{j=1}^{dk}f(x_j)$$ if and only if the mapping $f$ : X ${\rightarrow}$ Y is Cauchy additive, and prove the Cauchy-Rassias stability of the functional equation ($\ddag$) in Banach modules over a unital $C^{\ast}$-algebra. Let $\mathcal{A}$ and $\mathcal{B}$ be unital $C^{\ast}$-algebras. As an application, we show that every almost homomorphism $h\;:\;\mathcal{A}{\rightarrow}\mathcal{B}$ of $\mathcal{A}$ into $\mathcal{B}$ is a homomorphism when $h((k-1)^nuy)=h((k-1)^nu)h(y)$ for all unitaries $u{\in}\mathcal{A}$, all $y{\in}\mathcal{A}$, and $n$ = 0,1,2,$\cdots$. Moreover, we prove the Cauchy-Rassias stability of homomorphisms in $C^{\ast}$-algebras.