  We derive the Kac and new determinant formulae for an arbitrary (integer)
level $K$ fractional superconformal algebra using the BRST cohomology
techniques developed in conformal field theory. In particular, we reproduce the
Kac determinants for the Virasoro ($K=1$) and superconformal ($K=2$) algebras.
For $K\geq3$ there always exist modules where the Kac determinant factorizes
into a product of more fundamental new determinants. Using our results for
general $K$, we sketch the non-unitarity proof for the $SU(2)$ minimal series;
as expected, the only unitary models are those already known from the coset
construction. We apply the Kac determinant formulae for the spin-4/3
parafermion current algebra ({\em i.e.}, the $K=4$ fractional superconformal
algebra) to the recently constructed three-dimensional flat Minkowski
space-time representation of the spin-4/3 fractional superstring. We prove the
no-ghost theorem for the space-time bosonic sector of this theory; that is, its
physical spectrum is free of negative-norm states.
