Jordan, Jordan Right and Jordan Left Derivations on Convolution Algebras

Jordan derivations on the $θ-$Lau products of Banach algebras

“…Let d(a, 0) = (x 0 , y 0 ) and d(ua, 0) = (x 1 , y 1 ) for some x 0 , x 1 ∈ A and y 0 , y 1 ∈ B. If we replace a by ua in (1), then (x 1 + x 1 , y 1 + y 1 ) = (x 1 + φ(y 1 )u + za + γ(y 1 )u + uaz + ux 1 , 0), Hence y 1 = 0 and by (1), y 0 = 0. Therefore d maps A into itself.…”

“…Here a question arises: when dose the converse hold? In 1957, Herstein [10] proved that every Jordan derivation on a 2-torsion free prime ring is a derivation; see also [1,5,14,15]. Bresar [4] gave a generalization of Herstein's result for semiprime rings.…”

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“…Then L ∞ (G) * and L ∞ 0 (G) * are Banach algebras with the first Arens product. One can prove that L ∞ (G) * and L ∞ 0 (G) * have right identities [5,9]; for more study see [1,2,[11][12][13][14]. Let M(G) be the measure algebra of G. Then M(G) with the convolution product is a unital Banach algebra and M(G) ∼ = C 0 (G) * , where C 0 (G) is the space of all complexvalued continuous functions on G that vanish at infinity [7].…”

“…Cusack [5] extended this result to semiprime rings. Sinclair [13] proved that every continuous Jordan derivation of a semisimple Banach algebra is a derivation; for derivations and Jordan derivations on group algebras see [1,2,12]. Jordan left derivations have been introduced and studied by Brešar and Vukman [4].…”

Symmetric bi-derivations and their generalizations on group algebras

“…This shows that ∆ m is a Jordan derivation of L ∞ 0 (G) * for all m ∈ L ∞ 0 (G) * . By [1] every Jordan derivation of L ∞ 0 (G) * is a derivation of L ∞ 0 (G) * . Hence ∆ m is a derivation of L ∞ 0 (G) * .…”