  Let A and B be two commutative semisimple Frechet algebras. We first give a
characterization of the multiplier algebra of the direct sum of A and B. We
then prove that A \oplus B is a BSE-algebra if and only if A and B are
BSE-algebras. Furthermore, for a closed ideal I of A, we study multipliers of
ideals and quotient algebras of A and show that I and A/I are BSE-algebras,
under certain conditions.
