  An extended Bose-Einstein condensate (BEC) in an optical lattice provides a
kind of periodic dielectric and causes band gaps to occur in the spectrum of
light propagating through it. We examine the question whether these band gaps
can modify the spontaneous emission rate of atoms excited from the BEC, and
whether they can lead to a self-stabilization of the BEC against spontaneous
emission. We find that self-stabilization is not possible for BECs with a
density in the order of $10^{14}$ cm$^{-3}$. However, the corresponding
non-Markovian behavior produces significant effects in the decay of excited
atoms even for a homogeneous BEC interacting with a weak laser beam. These
effects are caused by the occurrence of an avoided crossing in the photon (or
rather polariton) spectrum. We also predict a new channel for spontaneous decay
which arises from an interference between periodically excited atoms and
periodic photon modes. This new channel should also occur in ordinary periodic
dielectrics.
