An elementary proof for a characterization of *-isomorphisms

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 Hilbert 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.