  In this paper we prove an abstract homomorphism theorem for bilinear
multipliers in the setting of locally compact Abelian (LCA) groups. We also
provide some applications. In particular, we obtain a bilinear abstract version
of K. de Leeuw's theorem for bilinear multipliers of strong and weak type. We
also obtain necessary conditions on bilinear multipliers on non-compact LCA
groups, yielding boundedness for the corresponding operators on products of
rearrangement invariant spaces. Our investigations extend some existing results
in Euclidean spaces to the framework of general LCA groups, and yield new
boundedness results for bilinear multipliers in quasi Banach spaces.

Functional Analysis