Operator space structure on Feichtinger’s Segal algebra

We extend the definition, from the class of abelian groups to a general locally compact group G, of Feichtinger’s remarkable Segal algebra S0(G). In order to obtain functorial properties for non-abelian groups, in particular a tensor product formula, we endow S0(G) with an operator space structure. With this structure S0(G) is simultaneously an operator Segal algebra of the Fourier algebra A(G), and of the group algebra L1(G). We show that this operator space structure is consistent with the major functorial properties: (i) S0(G)⊗̂S0(H) ∼= S0(G×H) completely isomorphically (operator projective tensor product), if H is another locally compact group; (ii) the restriction map u 7→ u|H : S0(G) → S0(H) is completely surjective, if H is a closed subgroup; and (iii) τN : S0(G) → S0(G/N) is completely surjective, where N is a normal subgroup and τNu(sN) = ∫ N u(sn)dn. We also show that S0(G) is an invariant for G when it is treated simultaneously as a pointwise algebra and a convolutive algebra.