ON PROJECTIONS OF L"(G) ONTO TRANSLATION-INVARIANT SUBSPACES By JOHN E. GILBERT [Received 19 December 1966—Revised 27 September 1967] 1. For a locally compact abelian group 0 having character group F let T be an invariant subspace of L m (G) } i.e. a weak*-closed, translation- invariant subspace. In this paper interest centres on finding conditions for the existence of a projection (a bounded linear idempotent operator) of L^G) onto T. It is well known, for instance, that when G is the circle group T there is no projection from L°°(T) onto H m {T) (Hoffman ((8) 155)), in contrast to the Riesz projection theorem for L V (T), 1 < p < oo. On the other hand, one consequence of the theory of idempotent measures on G (Cohen (2)) is that, when G is compact, there is a projection of L°°(G) onto T if and only if the spectrum of r belongs to the coset-ring 2(F) of F. The principal result we prove is THEOREM A. Let Q. be a closed subset of F, r a an invariant subspace of L CO (G) with sp(r n ) = Q. Then there is a projection P from L CO (G) onto r n if and only if Cl belongs to the coset-ring 2 d (F) of the discrete dual of G. The projection can be chosen so that it commutes with convolution with functions in L X (G), i.e. P(f*<p)=f*P( ? ), <peL<°(G), fzL\G). (1) The necessity of this condition is known already (Rosenthal (14)). The proof of sufficiency breaks into three quite separate steps. First, after a long but basically simple combinatorial-topological argument we describe all closed subsets of F which belong to 2 d (F) (Theorem 3.1). Any such set is a finite union of sets of the form A(II\A), where A e F, II is a closed subgroup of F, and A belongs to the coset-ring 2(11) of II. Very easily it is shown that all such sets are Ditkin sets, in particular sets of spectral synthesis (Theorem 3.9). Finally, an induction argument is used to construct the required projection onto r a once a projection has been constructed for the special case when Q = A(II\A). For this the existence of invariant means on ^ U (H) (bounded, uniformly continuous functions on a locally compact abelian group H) plays a crucial role. Many of the results of the paper can be carried over to non-abelian groups having invariant means, but any such consideration is reserved for a future paper. Proc. London Math. Soc. (3) 19 (1969) 69-88