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Superstability of an Exponential Equation in \({{C^*}}\)-Algebras

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Abstract

The aim of this paper is to prove the superstability of the following functional equations

$$f\left(\frac{x+y}{m}\right)^m \,=\, g(x)h(y),$$

where \({f, g, h: V^{2} \to A}\) are unknown mappings and m is a fixed positive integer. Here V is a vector space, and A is a unital normed algebra.

Furthermore, we prove the superstability of the following generalized Pexider exponential equation

$$f\left(\frac{x+y}{r}\right)^r = g(x)h(y),$$

where \({f, g, h: V^{2}\to I(A)\cap A^+}\) are unknown mappings and r is a fixed nonzero rational number. Here V is a vector space, I(A) is the set of all invertible elements in a commutative unital C *-algebra A and \({A^+}\) is the positive cone of A.

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Authors and Affiliations

  1. Department of Mathematics, Kangnam University, Yongin, Gyeonggi, 446-702, Republic of Korea

    Gwang Hui Kim

  2. Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, Republic of Korea

    Choonkil Park

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  1. Gwang Hui Kim
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  2. Choonkil Park
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Correspondence to Gwang Hui Kim.

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Kim, G.H., Park, C. Superstability of an Exponential Equation in \({{C^*}}\)-Algebras. Results. Math. 67, 197–205 (2015). https://doi.org/10.1007/s00025-014-0404-4

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