Properties of Three Algebras Related to ^ - Multipliers

A,(T) = [L9(T) ® LQ(T)]IK where K is the kernel of the convolution operator c:Lp(S>LQ(T)->C0(T) by (f®g)(y)(f* g)(y)-4*(r) is the /?-Fourier algebra which was introduced by Figa-Talamanca in [6] where it was shown that ^ ( T ) * isisometrically isomorphic to MP(T), the bounded, translation invariant, linear operators on LP(T), Herz [11] showed that AP(V) is a Banach algebra under pointwise multiplication; it is known that A2(r)=A(r)=L1(G) and that the inclusions A2(T)<^Ap(T)czA1(r)=C0(r) are norm decreasing if l</?<2; see [5], [6], [11] for the basic properties of AP(T). Let BP(T) denote the algebra of continuous functions ƒ on T such that f(y)h(y) eAv(T) whenever h eAP(T). The multiplier algebra BP(T) is a Banach algebra in the operator norm. We have studied BP(F) in [8], [9]. Fix/? in l<p<2. Regard L^T) as an algebra of convolution operators on LP(T) and let mp(T) denote the closure of L^V) in MP(T). The first result of this paper says that BV(T) is isometrically isomorphic to the dual space ^ ( T ) * . In the second result, we use certain properties of BP(V) to give a theorem of Eberlein type for ^ ( T ) . In the final section of the paper, we use