Uniqueness of translation invariant norms

Let A be a Banach function space and let M be a family of multipliers on A. We provide conditions on M so that the original topology of A is the only complete norm topology on A making all of the maps from M continuous. As a corollary we show that for a compact abelian group G, and a circle group T for A = L-P(T)(-), 1 < p < infinity, the L-P-norm is the only one that makes all trans rations continuous, while for A = C(G), A = L-infinity(G), or A = L-1(G) there are other norms with that property. For noncompact groups the situation is different-on the space L-1(R) the L-1-norm is the only one that makes a single nontrivial translation continuous. (C) 2000 Academic Press.