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Let G be a locally compact group, A(G) its Fourier algebra and L1(G) thespace of Haar integrable functions on G. We study the Segal algebraSA(G)=A(G)\cap L1(G) in A(G). It admits an operator space structure which makesit a completely contractive Banach algebra. We compute the dual space of SA(G).We use it show that restriction operator u|->u|H:SA(G)->A(H), for some non-openclosed subgroups H, is a surjective complete qutient map. We also show that ifN is a non-compact closed subgroup, then the averaging operatortau_N:SA(G)->L1(G/N), tau_N u(sN)=\int_N u(sn)dn is a surjective completequotient map. This puts an operator space perspective on the philosophy thatSA(G) is ``locally A(G) while globally L1''. Also, using the operator spacestructure we can show that SA(G) is operator amenable exactly when when G iscompact; and we can show that it is always operator weakly amenable. To obtainthe latter fact, we use E. Samei's theory of hyper-Tauberian Banach algebras.

Operator Segal algebras in Fourier algebras