LIMITS OF PURE FUNCTIONALS OF C*-ALGEBRAS

It is shown that, for a C*-algebra A, every (weak*-) limit of pure functionals is a multiple of a pure functional if and only if every limit of pure states is a multiple of pure states (a condition previously studied by Glimm). On the other hand, it is shown that the set of pure states P(A) being closed does not force the set of pure functionals <(G(A))over bar> to be closed. The conditions G(A) = G(A) and <(G(A))over bar> = G(A) U {0} are characterised in terms of sums of homogeneous C*-algebras.