We characterize C∗-algebras
with continuous trace among all C∗-algebras by a condition on the set P(A) of pure
states. The condition is that (1) the graph R(A) of the unitary equivalence relation
on P(A) is closed in P(A) ×P(A), and (2) transition probabilities are continuous for
the product topology on R(A) (i.e. that inherited from P(A) × P(A)). If R(A) is
given the quotient topology, these conditions are equivalent to properness of
the inclusion map from R(A) into P(A) × P(A). We show the product and
quotient topologies on R(A) coincide iff transition probabilities are continuous
for the product topology, and this in turn is equivalent to Fell’s condition.
Transition probabilities are always continuous for the quotient topology on
R(A).
