  We study the Bose-Einstein condensation (BEC) for a system of $^7Li$ atoms,
which have negative scattering length (attractive interaction), confined in a
harmonic potential. Within the Bogoliubov and Popov approximations, we
numerically calculate the density profile for both condensate and
non-condensate fractions and the spectrum of elementary excitations. In
particular, we analyze the temperature and number-of-boson dependence of these
quantities and evaluate the BEC transition temperature $T_{BEC}$. We calculate
the loss rate for inelastic two- and three-body collisions. We find that the
total loss rate is strongly dependent on the density profile of the condensate,
but this density profile does not appreciably change by increasing the thermal
fraction. Moreover, we study, using the quasi-classical Popov approximation,
the temperature dependence of the critical number $N_c$ of condensed bosons,
for which there is the collapse of the condensate. There are different regimes
as a function of the total number $N$ of atoms. For $N<N_c$ the condensate is
always metastable but for $N>N_c$ the condensate is metastable only for
temperatures that exceed a critical value $T_c$.
