p-OPERATOR SPACE STRUCTURE ON FEICHTINGER-FIGA-TALAMANCA-HERZ SEGAL ALGEBRAS

Abstract

We consider the minimal boundedly-translation-invariant Segal algebra S-0(p)(G) in the Figa-Talamanca-Herz algebra A(p) (G) of a locally compact group G. In the case that p = 2 and G is abelian this is the classical Segal algebra of Feichtinger. Hence we call this the Feichtinger-Figa-Talamanca-Herz Segal algebra of G. This space is also a Segal algebra in L-1(G) and is, remarkably, the minimal such algebra which is closed under pointwise multiplication by A(p) (G). Even for p = 2, this result is new for non-abelian G. We place a p-operator space structure on S-0(p)(G) based on work of Daws (M. DAWS, J. Operator Theory 63(2010), 47-83) and demonstrate the naturality of this by showing that it satisfies all natural functorial properties: projective tensor products, restriction to subgroups and averaging over normal subgroups. However, due to complications arising within the theory of p-operator spaces, we are forced to work with weakly complete quotient maps and weakly complete surjections (a class of maps we define).