We give an elementary proof of a result which characterizes onto *-isomorphisms of the algebra BL(H) of all the bounded linear operators on a Hubert space H. A known proof of this result (Arveson, 1976) relies on the theory of irreducible representations of C*-algebras, whereas the proof given by us is based on elementary properties of operators on a Hilbert space which can be found in any introductory text on Functional Analysis. © 2005 American Mathematical Society.