ISOMORPHISMS IN UNITAL C*-ALGEBRAS

It is shown that every almost linear bijection h : a → b of a unital c*-algebra a onto a unital c*-algebra b is a c*-algebra isomorphism when h(3nuy) = h(3nu)h(y) for all unitaries u ϵ a, all y ϵ a, and all n ϵ z, and that almost linear continuous bijection h : a → b of a unital c*-algebra a of real rank zero onto a unital c*-algebra b is a c*-algebra isomorphism when h(3nuy) = h(3nu)h(y) for all u ϵ {v ϵ a | v = v*, kvk = 1, v is invertible}, all y ϵ a, and all n ϵ z. assume that x and y are left normed modules over a unital c*-algebra a. it is shown that every surjective isometry t : x → y , satisfying t(0) = 0 and t(ux) = ut(x) for all x ϵ x and all unitaries u ϵ a, is an a-linear isomorphism. this is applied to investigate c*-algebra isomorphisms in unital c*-algebras.