Operator Segal algebras in Fourier algebras

Brian E. Forrest, Nico Spronk, and Peter J. Wood. "Operator Segal algebras in Fourier algebras." Studia Mathematica 179.3 (2007): 277-295. <http://eudml.org/doc/286496>.

@article{BrianE2007,
abstract = {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.},
author = {Brian E. Forrest, Nico Spronk, Peter J. Wood},
journal = {Studia Mathematica},
keywords = {operator Segal algebra; Fourier algebra},
language = {eng},
number = {3},
pages = {277-295},
title = {Operator Segal algebras in Fourier algebras},
url = {http://eudml.org/doc/286496},
volume = {179},
year = {2007},
}

TY - JOUR
AU - Brian E. Forrest
AU - Nico Spronk
AU - Peter J. Wood
TI - Operator Segal algebras in Fourier algebras
JO - Studia Mathematica
PY - 2007
VL - 179
IS - 3
SP - 277
EP - 295
AB - 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.
LA - eng
KW - operator Segal algebra; Fourier algebra
UR - http://eudml.org/doc/286496
ER -