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=Lp(T), 1<p<∞, the Lp-norm is the only one that makes all translations continuous, while • for A=C(G), A=L∞(G), or A=L1(G) there are other norms with that property. For noncompact groups the situation is different—on the space L1(R) the L1-norm is the only one that makes a single nontrivial translation continuous.