  For every $p\geq 2$, we give a characterization of positive definite
functions on a free group with finitely many generators, which can be extended
to the positive linear functionals on the free group $C^*$-algebra associated
with the ideal $\ell_p$. This is a generalization of Haagerup's
characterization for the case of the reduced free group $C^*$-algebra. As a
consequence, the associated $C^*$-algebras are mutually non-isomorphic, and
they have a unique tracial state.
