Operator Segal algebras in Fourier algebras

Let G be a locally compact group, A(G) its Fourier algebra and L-1 (G) the space of Haar integrable functions on G. We study the Segal algebra S(1)A(G) = A(G) boolean AND L-1(G) in A(G). It admits an operator space structure which makes it a completely contractive Banach algebra. We compute the dual space of S(1)A(G). We use it to show that the restriction operator u vertical bar -> u vertical bar(H) : S(1)A(G) -> A(H), for some non-open closed subgroups H, is a surjective complete quotient map. We also show that if N is a non-compact closed subgroup, then the averaging operator tau(N) : S(1)A(G) -> L-1(G/N), tau(N)u(sN) = integral(N) u(sn) dn, is a surjective complete quotient map. This puts an operator space perspective on the philosophy that S(1)A(G) is "locally A(G) while globally L-1". Also, using the operator space structure we can show that S1A(G) is operator amenable exactly when when G is compact; and we can show that it is always operator weakly amenable. To obtain the latter fact, we use E. Samei's theory of hyper-Tauberian Banach algebras.