Operator space structure on Feichtinger's Segal algebra

Abstract

We extend the definition, from the class of abelian groups to a general locally compact group G, of Feichtinger's remarkable Segal algebra S_0(G). In order to obtain functorial properties for non-abelain groups, in particular a tensor product formula, we endow S_0(G) with an operator space structure. With this structure S_0(G) is simultaneously an operator Segal algebra of the Fourier algebra A(G), and of the group algebra L^1(G). We show that this operator space structure is consistent with the major functorial properties: (i) S_0(G)\hat{\otimes}S_0(H)=S_0(G\times H) completely isomorphically (operator projective tensor product), if H is another locally compact group; (ii) the restriction map u\mapsto u|_H:S_0(G)\to S_0(H) is completely surjective, if H is a closed subgroup; and (iii) T_N:S_0(G)\to S_0(G/N) is completely surjective, where N is a normal subgroup and T_N u(sN)=\int_N u(sn)dn. We also show that S_0(G) is an invariant for G when it is treated simultaneously as a pointwise algebra and a convolutive algebra.