On trigonometric series associated with separable, translation invariant subspaces of $L^{\infty }(G)$

Abstract:

$G$ denotes a compact abelian group, and $\Gamma$ denotes its dual. Our main result is that every non-Sidon set $E \subset \Gamma$ contains a non-Sidon set $F$ such that $L_F^\infty (G) = { \oplus _l}1_{i = 1}^\infty {C_{{F_i}}}(G)$, where the ${F_i}$’s are finite, mutually disjoint, and $\cup _{i = 1}^\infty {F_i} = F$.