Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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QUADRATIC FUNCTIONAL EQUATIONS ASSOCIATED WITH BOREL FUNCTIONS AND MODULE ACTIONS

  • Published : 2009.05.31
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Abstract

For a Borel function ${\psi}:\mathbb{R}{\times}\mathbb{R}{\rightarrow}\mathbb{R}$ satisfying the functional equation $\psi$ (s + t, u + v) + $\psi$(s - t, u - v) = $2\psi$(s, u) + $2\psi$(t, v), we show that it satisfies the functional equation $$\psi$$(s, t) = s(s - t)$$\psi$$(1, 0) + $$st\psi$$(1, 1) + t(t - s)$$\psi$$(0, 1). Using this, we prove the stability of the functional equation f(ax + ay, bz + bw) + f(ax - ay, bz - bw) = 2abf(x, z) + 2abf(y,w) in Banach modules over a unital $C^*$-algebra.

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References

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