  We describe the ghost sector of cubic string field theory in terms of degrees
of freedom on the two halves of a split string. In particular, we represent a
class of pure ghost BRST operators as operators on the space of half-string
functionals. These BRST operators were postulated by Rastelli, Sen, and
Zwiebach to give a description of cubic string field theory in the closed
string vacuum arising from condensation of a D25-brane in the original
tachyonic theory. We find a class of solutions for the ghost equations of
motion using the pure ghost BRST operators. We find a vanishing action for
these solutions, and discuss possible interpretations of this result. The form
of the solutions we find in the pure ghost theory suggests an analogous class
of solutions in the original theory on the D25-brane with BRST operator Q_B
coupling the matter and ghost sectors.
