Functional calculus in weighted group algebras

is converging for all s and such that‖ϕ{f s } ∗ F − F ‖ ω → 0for all continuous functions F with compact support in G. Moreover the functions ϕ{f s }are contained in m(∅) by construction.In fact, let’s choose ϕ ∈ Φ such that ϕ(1) = 1. Then‖ϕ{f s } ∗ F − F ‖ ω = ‖ ∑ n∈Zˆϕ(n)[e inf s∗ F − e in F ]‖ ω .As the functions f s are uniformly bounded in L 1 (G, ω), it is easy to check that for everyfixed n,e inf s∗ F → e in Fin L 1 (G, ω) as s → ∞. So, for any fixed N ∈ N, we have∑|n|≤Nˆϕ(n)[e inf s∗ F − e in F ] → 0in L 1 (G, ω). Next we have to show that we may choose ϕ such that, for any ε > 0, thereexists N ∈ N such that‖ ∑ˆϕ(n)[e inf s∗ F − e in F ]‖ ω ≤ ∑∑| ˆϕ(n)|‖e inf s∗ F ‖ ω + | ˆϕ(n)e in |‖F ‖ ω|n|>N< ε,|n|>N|n|>Nindependently of s. Suppose that we had already determined ϕ and N 1 such that∑|n|>N 1| ˆϕ(n)|‖e inf s∗ F ‖ ω < ε 2 ,for all s, then (as ∑ n∈Z ˆϕ(n)ein = ϕ(1) = 1 converges) we can choose N ≥ N 1 such that|∑|n|>Nˆϕ(n)e in |‖F ‖ ω < ε 2 .Thus it suffices to show (∗). According to (3.12.)(∗)‖e inf s∗ F ‖ ω ≤ K 2 (1 + |n|)(1 + |n| 2 ) Q 2 s(|n|)2 ln(ln |n|) e K 3(with constants K 2 and K 3 given byK 2 = 2 Q 2 K12 (1 + q)Q2 s(q)‖F ‖2 + ‖F ‖ ωK 3 = 2e (eC′p )‖f s ‖ ω,33|n|(ln |n|) C )