  We investigate the set of maximally mixed states of a C*-algebra, extending
previous work by Alberti on von Neumann algebras. We show that, unlike for von
Neumann algebras, the set of maximally mixed states of a C*-algebra may fail to
be weak* closed. We obtain, however, a concrete description of the weak*
closure of this set, in terms of tracial states and states which factor through
simple traceless quotients. For C*-algebras with the Dixmier property or with
Hausdorff primitive spectrum we are able to advance our investigations further.
In the latter case we obtain a concrete description of the set of maximally
mixed states in terms of traces and extensions of the states of a closed
two-sided ideal. We pose several questions.
