Operator Segal algebras in Fourier algebras

Brian E. Forrest ; Nico Spronk ; Peter J. Wood

Studia Mathematica, Tome 178 (2007), p. 277-295 / Harvested from The Polish Digital Mathematics Library

Résumé

Let G be a locally compact group, A(G) its Fourier algebra and L¹(G) the space of Haar integrable functions on G. We study the Segal algebra S¹A(G) = A(G) ∩ L¹(G) in A(G). It admits an operator space structure which makes it a completely contractive Banach algebra. We compute the dual space of S¹A(G). We use it to show that the restriction operator $u ↦ u|_{H}: S¹A(G) → A(H)$, for some non-open closed subgroups H, is a surjective complete quotient map. We also show that if N is a non-compact closed subgroup, then the averaging operator $τ_{N}: S¹A(G) → L¹(G/N)$, $τ_{N}u(sN) = ∫_{N} u(sn)dn$, is a surjective complete quotient map. This puts an operator space perspective on the philosophy that S¹A(G) is “locally A(G) while globally L¹”. Also, using the operator space structure we can show that S¹A(G) is operator amenable exactly when when G is compact; and we can show that it is always operator weakly amenable. To obtain the latter fact, we use E. Samei’s theory of hyper-Tauberian Banach algebras.

Détail

Publié le : 2007-01-01

Zbl 1112.43003

EUDML-ID : urn:eudml:doc:286496