Some Thin Sets in Discrete Abelian Groups 57

Let T be a discrete abelian group, and E C T. For F C E, we say that F e 9(E), if for all A, finite subsets of I", 0 / A, A + F n F is finite. Having defined the Banach algebra, A(E) = c(E) n B(E), we prove the following: (i) E C T is a Sidon set if and only if every F e 9(E) is a Sidon set; (ii) E e?(r) is a Sidon set if and only if A(E) = A(E). 0. Introduction. Let L denote a discrete abelian group, and let G denote its dual. We let G , denote the abelian group G endowed with the discrete topology; (Gj) = T is the Bohr compactification of I\ V is dense in V, and C(D is identified naturally with the almost periodic functions on V* In what follows below, we use standard notation and facts as presented in Chapters 1 and 2 of [9]. For £Cr, set A(F)=L1(G)7{/e LX(G):/= 0 on E\ and B(E) = M(G)7l/i e M(G) : p. = 0 on E\, where the quotients are the usual Banach algebra quotients. Let c(E) denote all bounded functions on E which vanish at infinity. Clearly, A(E) C c(E), and B(E) C l°°(E); A(E) is norm dense in c(E), and ||g||B(ß) > llglL for all g £ B(E). We set A(E) = c(E) n B(E). Equipped with the B(E)-norm, and pointwise multiplication A(E) is a Banach algebra. Since (CE(G))* = B(E), and (A(E))* = LTM(G), A(E) is isometrically imbedded in B(E) and, therefore, in A(E). (If R(G) is any subspace of L (G), Re(G) denotes all functions in R(G) whose spectrum is in E-.) Detailed studies of algebras related to A(E) appear in [5], [7], [8], [ll], and [12]. When A(E) = c(E), we say that E is a Sidon set, and define the Sidon conReceived by the editors June 5, 1972 and, in revised form, April 2, 1973. AMS (MOS) subject classifications (1970). Primary 43A25.