Operator Segal algebras in Fourier algebras

• Autorzy: Brian E. Forrest , Nico Spronk , Peter J. Wood
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
• 46L07: Operator spaces and completely bounded maps
• 43A07: Means on groups, semigroups, etc.; amenable groups
• 47L25: Operator spaces (= matricially normed spaces)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2007
• Tom: 179
• Numer: 3
• Strony: 277-295

Daty

wydano

2007

Twórcy

autor

Brian E. Forrest

• Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

autor

Nico Spronk

• Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

autor

Peter J. Wood

• Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

Identyfikatory

DOI

10.4064/sm179-3-5