Translation-invariant subspaces of functions on the group of linear transformations on the real line

The characterization of right translation-invariant subspaces of L _∞(G ^*), where [InlineMediaObject not available: see fulltext.], is studied. We introduce the class of multiplier functions which, in the semisimple case, play a role similar to that played by the exponentials for the real line. However, it is proved that multiplier functions of G ^* with respect to R fail to characterize right translation-invariant subspaces of L _∞(G ^*). That is, we construct a right translation-invariant, w^*-closed subspace of L _∞(G ^*) which contains no multiplier function.