proceedings of the american mathematical society Volume 34, Number 2, August 1972 ¿"-CONVOLUTION OPERATORS SUPPORTED BY SUBGROUPS1 CHARLES F. DUNKL AND DONALD E. RAMIREZ Abstract. Let G be a compact nonabelian group and H be a closed subgroup of G. Then H is a set of spectral synthesis for the Fourier algebra A(G) (and indeed for A"{G), \^p<<x>). For l^/j<co, each L"(G)-multiplier T corresponds to a Z."(//^multi- plier S by the rule {Tf)\H=S{ f\H), fe A{G),if and only if the support of T is contained in H. Let G be a compact nonabelian group and G its dual. We denote the Fourier algebra by A(G) and its dual by ^œ(G). We will use the notation from our book [1]. Let </> e ¿¡¿""(G), then the support of <j>, denoted by spt <f>, is defined to be the intersection of the sets {K^G:Kis compact and(/, </S)=0 whenever the support of fcG\K,fe A(G)} [1, p. 94]. For/eC(G), spt/denotes the usual support of/. For u a bounded Borel function on G, define ü by ú(x) = u(x~1), x eG. Let F be a closed subset of G. The set E is called a set of spectral synthesis for A(G) provided whenever fe A(G), f(x)=0 for xeE, and £>0, there exists g e A(G) with g=0 on a neighborhood of E and ||/— g\\A<e. We will show that closed subgroups H of G are sets of spectral synthesis for A(G). Our proof is adapted from [3] where the result is given for H normal. Henceforth H will be a fixed closed subgroup of G, with normalized Haar measure mH. Proposition 1. Let fe A(G), /=0 on H, and e>0. Then there exists a neighborhood W of the identity e of G such that if u is a nonnegative bounded Borel function on W, and JG u(x) dmG(x)=l, then Proof. Since translation is continuous in A(G) [1, p. 91], there exists a neighborhood W of e such that if y e W, then \\f— R(y)f\\A^e (R(y)f(x)=f(xy),x,yeG). Received by the editors October 12, 1971. AMS 1970 subject classifications. Primary 43A22, 43A45. Key words and phrases. Fourier algebra, spectral synthesis, L "-multiplier. 1 This research was supported in part by NSF contract no. GP-19852. © American Mathematical Society 1972 475 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use