Operator Figà-Talamanca-Herz algebras

Volker Runde. "Operator Figà-Talamanca-Herz algebras." Studia Mathematica 155.2 (2003): 153-170. <http://eudml.org/doc/285290>.

@article{VolkerRunde2003,
abstract = {Let G be a locally compact group. We use the canonical operator space structure on the spaces $L^\{p\}(G)$ for p ∈ [1,∞] introduced by G. Pisier to define operator space analogues $OA_\{p\}(G)$ of the classical Figà-Talamanca-Herz algebras $A_\{p\}(G)$. If p ∈ (1,∞) is arbitrary, then $A_\{p\}(G) ⊂ OA_\{p\}(G)$ and the inclusion is a contraction; if p = 2, then OA₂(G) ≅ A(G) as Banach spaces, but not necessarily as operator spaces. We show that $OA_\{p\}(G)$ is a completely contractive Banach algebra for each p ∈ (1,∞), and that $OA_\{q\}(G) ⊂ OA_\{p\}(G)$ completely contractively for amenable G if 1 < p ≤ q ≤ 2 or 2 ≤ q ≤ p < ∞. Finally, we characterize the amenability of G through the existence of a bounded approximate identity in $OA_\{p\}(G)$ for one (or equivalently for all) p ∈ (1,∞).},
author = {Volker Runde},
journal = {Studia Mathematica},
keywords = {(operator) amenability; complex interpolation; (operator) Figà-Talamanca-Herz algebra; Fourier algebra; locally compact group; operator space; operator -space; completely contractive Banach algebra},
language = {eng},
number = {2},
pages = {153-170},
title = {Operator Figà-Talamanca-Herz algebras},
url = {http://eudml.org/doc/285290},
volume = {155},
year = {2003},
}

TY - JOUR
AU - Volker Runde
TI - Operator Figà-Talamanca-Herz algebras
JO - Studia Mathematica
PY - 2003
VL - 155
IS - 2
SP - 153
EP - 170
AB - Let G be a locally compact group. We use the canonical operator space structure on the spaces $L^{p}(G)$ for p ∈ [1,∞] introduced by G. Pisier to define operator space analogues $OA_{p}(G)$ of the classical Figà-Talamanca-Herz algebras $A_{p}(G)$. If p ∈ (1,∞) is arbitrary, then $A_{p}(G) ⊂ OA_{p}(G)$ and the inclusion is a contraction; if p = 2, then OA₂(G) ≅ A(G) as Banach spaces, but not necessarily as operator spaces. We show that $OA_{p}(G)$ is a completely contractive Banach algebra for each p ∈ (1,∞), and that $OA_{q}(G) ⊂ OA_{p}(G)$ completely contractively for amenable G if 1 < p ≤ q ≤ 2 or 2 ≤ q ≤ p < ∞. Finally, we characterize the amenability of G through the existence of a bounded approximate identity in $OA_{p}(G)$ for one (or equivalently for all) p ∈ (1,∞).
LA - eng
KW - (operator) amenability; complex interpolation; (operator) Figà-Talamanca-Herz algebra; Fourier algebra; locally compact group; operator space; operator -space; completely contractive Banach algebra
UR - http://eudml.org/doc/285290
ER -