  The functor Max of Mulvey assigns to each unital C*-algebra A the unital
involutive quantale Max A of closed linear subspaces of A, and it has been
remarked that it classifies unital C*-algebras up to *-isomorphism. In this
paper we provide a proof of this and of the stronger fact that for every
isomorphism u : Max A -> Max B of unital involutive quantales there is a
*-isomorphism u' : A -> B such that Max u' coincides with u when restricted to
the left-sided elements of Max A. But we also show that isomorphisms u : Max A
-> Max B may exist for which no isomorphism v : A -> B is such that Max v = u.
