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-abelian 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 ) ⊗ ˆ S 0 ( H ) ≅ S 0 ( G × H ) completely isomorphically (operator projective tensor product), if H is another locally compact group; (ii) the restriction map u ↦ u | H : S 0 ( G ) → S 0 ( H ) is completely surjective, if H is a closed subgroup; and (iii) τ N : S 0 ( G ) → S 0 ( G / N ) is completely surjective, where N is a normal subgroup and τ N u ( s N ) = ∫ N u ( s n ) d n . 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.