2017/02/21 by Varvara Shepelska, Yong Zhang, Shepelska, Varvara +1
Mathematics · #22D15 #43A10 (Secondary) #43A20 (Primary) #46H10 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Functional Analysis (math.FA)
paper · pdf · doi:10.48550/arxiv.1702.06605
openalex publication_date 2017/02/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Weak amenability of a weighted group algebra, or a Beurling algebra, is a long-standing open problem. The commutative case has been extensively investigated and fully characterized. We study the non-commutative case. Given a weight function ω on a locally compact group G, we characterize derivations from L1(G,ω) into its dual in terms of certain functions. Then we show that for a locally compact IN group G, if there is a non-zero continuous group homomorphism φ: G→ ℂ such that φ(x)/ω(x)ω(x-1) is bounded on G, then L1(G,ω) is not weakly amenable. Some useful criteria that rule out weak amenability of L1(G,ω) are established. Using them we show that for many polynomial type weights the weighted Heisenberg group algebra is not weakly amenable, neither is the weighted \boldsymbolax+b group algebra. We further study weighted quotient group algebra L1(G/H,ω), where ω is the canonical weight on G/H induced by ω. We reveal that the kernel of the canonical homomorphism from L1(G,ω) to L1(G/H,ω) is complemented. This allows us to obtain some sufficient conditions under which L1(G/H,ω) inherits weak amenability of L1(G,ω). We study further weak amenability of Beurling algebras of subgroups. In general, weak amenability of a Beurling algebra does not pass to the Beurling algebra of a subgroup. However, in some circumstances this inheritance can happen. We also give an example to show that weak amenability of both L1(H,ω|H) and L1(G/H,ω) does not ensure weak amenability of L1(G,ω).