Weak amenability of weighted group algebras on some discrete groups

Varvara Shepelska. "Weak amenability of weighted group algebras on some discrete groups." Studia Mathematica 230.3 (2015): 189-214. <http://eudml.org/doc/285545>.

@article{VarvaraShepelska2015,
abstract = {Weak amenability of ℓ¹(G,ω) for commutative groups G was completely characterized by N. Gronbaek in 1989. In this paper, we study weak amenability of ℓ¹(G,ω) for two important non-commutative locally compact groups G: the free group ₂, which is non-amenable, and the amenable (ax + b)-group. We show that the condition that characterizes weak amenability of ℓ¹(G,ω) for commutative groups G remains necessary for the non-commutative case, but it is sufficient neither for ℓ¹(₂,ω) nor for ℓ¹((ax + b),ω) to be weakly amenable. We prove that for several important classes of weights ω the algebra ℓ¹(₂,ω) is weakly amenable if and only if the weight ω is diagonally bounded. In particular, the polynomial weight $ω_\{α\}(x) = (1 + |x|)^\{α\}$, where |x| denotes the length of the element x ∈ ₂ and α > 0, never makes $ℓ¹(₂,ω_\{α\})$ weakly amenable. We also study weak amenability of an Abelian algebra ℓ¹(ℤ²,ω). We give an example showing that weak amenability of ℓ¹(ℤ²,ω) does not necessarily imply weak amenability of $ℓ¹(ℤ,ω_\{i\})$, where $ω_\{i\}$ denotes the restriction of ω to the ith coordinate (i = 1,2). We also provide a simple procedure for verification whether ℓ¹(ℤ²,ω) is weakly amenable.},
author = {Varvara Shepelska},
journal = {Studia Mathematica},
keywords = {derivation; weak amenability; weight; locally compact group},
language = {eng},
number = {3},
pages = {189-214},
title = {Weak amenability of weighted group algebras on some discrete groups},
url = {http://eudml.org/doc/285545},
volume = {230},
year = {2015},
}

TY - JOUR
AU - Varvara Shepelska
TI - Weak amenability of weighted group algebras on some discrete groups
JO - Studia Mathematica
PY - 2015
VL - 230
IS - 3
SP - 189
EP - 214
AB - Weak amenability of ℓ¹(G,ω) for commutative groups G was completely characterized by N. Gronbaek in 1989. In this paper, we study weak amenability of ℓ¹(G,ω) for two important non-commutative locally compact groups G: the free group ₂, which is non-amenable, and the amenable (ax + b)-group. We show that the condition that characterizes weak amenability of ℓ¹(G,ω) for commutative groups G remains necessary for the non-commutative case, but it is sufficient neither for ℓ¹(₂,ω) nor for ℓ¹((ax + b),ω) to be weakly amenable. We prove that for several important classes of weights ω the algebra ℓ¹(₂,ω) is weakly amenable if and only if the weight ω is diagonally bounded. In particular, the polynomial weight $ω_{α}(x) = (1 + |x|)^{α}$, where |x| denotes the length of the element x ∈ ₂ and α > 0, never makes $ℓ¹(₂,ω_{α})$ weakly amenable. We also study weak amenability of an Abelian algebra ℓ¹(ℤ²,ω). We give an example showing that weak amenability of ℓ¹(ℤ²,ω) does not necessarily imply weak amenability of $ℓ¹(ℤ,ω_{i})$, where $ω_{i}$ denotes the restriction of ω to the ith coordinate (i = 1,2). We also provide a simple procedure for verification whether ℓ¹(ℤ²,ω) is weakly amenable.
LA - eng
KW - derivation; weak amenability; weight; locally compact group
UR - http://eudml.org/doc/285545
ER -