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Biderivations and bihomomorphisms in Banach algebras

Park Choonkil (Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, Republic of Korea)

In this paper, we solve the following bi-additive s-functional inequalities || f(x+y,z+w) + f(x+y,z-w)+f(x-y,z+w) + f (x-y,z-w)- 4f(x,z)||≤ ||s(2f(x+y,z-w)+ 2f(x-y,z + w)- 4f(x,z) + 4f(y,w)||(1) and ||2f(x+y,z-w) + 2f(x-y,z+w)-4f(x,z) + 4f(y,w)|| (2)≤ ||s(f(x+y,z+w)+ f(x+y,z-w) + f(x-y,z+w)+f(x-y,z-w)-4f(x,z))||, where s is a fixed nonzero complex number with |s| < 1. Moreover, we prove the Hyers-Ulam stability of biderivations and bihomomorphismsions in Banach algebras and unital C+-algebras, associated with the bi-additive s-functional inequalities (1) and (2).

Keywords: biderivation on C*-algebra, bihomomorphism in Banach algebra, Hyers-Ulam stability, bi-additive s-functional inequality