Operator space structure on Feichtinger's Segal algebra

摘要

We extend the definition, from the class of abelian groups to a general locally compact group G, of Feichtinger's remarkable Segal algebra SO(G). In order to obtain functorial properties for non-abelian groups, in particular a tensor product formula, we endow SO (G) with an operator space structure. With this structure SO (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) (circle times) over cap S-0(H) congruent to S-0(G x H) completely isomorphically (operator projective tensor product), if H is another locally compact group; (ii) the restriction map u -> u vertical bar(H) : S-0(G) -> S-0(H) is completely surjective, if H is a closed subgroup; and (iii) tau(N): S-0 (G) ->. S-0(GIN) is completely surjective, where N is a normal subgroup and tau(N)u(s N) = f(N) u(sn) dn. We also show that So (G) is an invariant for G when it is treated simultaneously as a pointwise algebra and a convolutive algebra. (c) 2007 Elsevier Inc. All rights reserved.