Homomorphism-derivation functional inequalities in C*-algebras

Abstract

In this paper, we introduce and solve the following additive-additive (s, t)-functional inequality

parallel to g (x + y) - g(x) - g(y)parallel to + parallel to 2h(x+y/2) - h(x) - h(y)parallel to (0,1)

<= parallel to s(2g(x+y/2) - g(x) - g(y)parallel to + parallel to t(h(x=y) - h(x) - h(y))parallel to,

where s and t are fixed nonzero complex numbers with vertical bar s vertical bar < 1 and vertical bar t vertical bar < 1. Furthermore, we investigate homomorphisms and derivations in complex Banach algebras and unital C*-algebras, associated to the additive-additive (s, t)-functional inequality (0.1) under some extra condition.

Moreover, we introduce and solve the following additive-additive (s, t)-functional inequality

parallel to g (x + y + z) - g(x) - g(y) g(z)parallel to + parallel to 3h (x + y +z/3) + h(x -2y + z) + h(x + y -2z) - 3h(x)parallel to

<= parallel to s(3g(x+y+z/3) - g(x) - g(y) - g(z)parallel to (0,2)

+ parallel to t (h(x +y +z) + h(x - 2y + z) + h(x + y -2z) -3h(x))parallel to.

where s and t are fixed nonzero complex numbers with vertical bar s vertical bar < 1 and vertical bar t vertical bar < 1. Furthermore, we investigate C*-ternary derivations and C*-ternary homomorphisms in C*-ternary algebras, associated to the additive-additive (s, t)-functional inequality (0.2) under some extra condition.