Isomorphisms between 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(2nuy)=h(2nu)h(y) for all unitaries u∈A, all y∈A, and n=0,1,2,…, 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(2nuy)=h(2nu)h(y) for all u∈{v∈A|v=v∗,‖v‖=1,vis invertible}, all y∈A, and n=0,1,2,…. 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∈X and all unitaries u∈A, is an A-linear isomorphism. This is applied to investigate C∗-algebra isomorphisms between unital C∗-algebras.