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

The characterization of right translation-invariant subspaces ofL _∞(G ^*), where , 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 ofG ^* with respect toR fail to characterize right translation-invariant subspaces ofL _∞(G ^*). That is, we construct a right translation-invariant, w^*-closed subspace ofL _∞(G ^*) which contains no multiplier function. This paper is a part of the author's Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor H. Furstenberg, to whom the author wishes to express his thanks for his helpful guidance, and valuable remarks.