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 2 A, all y 2 A, and all n 2 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 2 {v 2 A | v = v, kvk = 1, v is invertible}, all y 2 A, and all n 2 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 2 X and all unitaries u 2 A, is an A-linear isomorphism. This is applied to investigate C*-algebra isomorphisms in unital C*-algebras.