  In search of states with non-Abelian statistics, we explore the fractional
quantum Hall effect in a system of two-dimensional charge carrier holes. We
propose a new method of mapping states of holes confined to a finite width
quantum well in a perpendicular magnetic field to states in a spherical shell
geometry. This method provides single-particle hole states used in exact
diagonalization of systems with a small number of holes in the presence of
Coulomb interactions. An incompressible fractional quantum Hall state emerges
in a hole liquid at the half-filling of the ground state in a magnetic field in
the range of fields where single-hole states cross. This state has a negligible
overlap with the Halperin 331 state, but a significant overlap with the
Moore-Read Pfaffian state. Excited fractional quantum Hall states for small
systems have sizable overlap with non-Abelian excitations of the Moore-Read
Pfaffian state.
