Let us recall the following definition (see [6]):
Definition 7.1. LetAbe a Banach∗-algebra. We say thatAhas theWiener property (W) if for every proper closed two-sided ideal I of A, there exists a topologically irreducible∗-representationπofAsuch thatI⊂kerπ. IfAis of the formL1(G) for some locally compact groupG, we also say that the groupGhas the Wiener property.
Examples 7.2. (i) In [15] it is shown that if G is a connected, simply connected, nilpotent Lie group and ifωis a polynomial weight onG, thenL1(G, ω) has the Wiener property.
(ii) In [6] it is shown that ifGis a compactly generated, locally compact group with polynomial growth and if ω is a weight on Gthat is at most sub-exponential, then the algebraL1(G, ω) has the Wiener property.
(iii) For abelian groups, Domar ([5]) has shown thatL1(G, ω) has the Wiener prop- erty if P+∞
n=1
ln(ω(xn))
n2 < +∞ for all x ∈ G. This is in particular the case if P+∞
n=1 ln(s(n))
n2 < +∞. For G = R, Vretblad ([20]) even shows the converse, for a certain type of weights: If ω(x) = exp(π2|x|q(x)) with q decreasing on R+ and ω increasing on R+ and if L1(R, ω) has the Wiener property, then P+∞
n=1
ln(ω(nx))
n2 <+∞(we write nxinstead of xn as G=R) for every x∈ R (see the introduction for more details).
In this section we shall study the Wiener property for algebras of the formL1(G, ω), whereGis a compactly generated, locally compact group with polynomial growth and ω is a weight on Gsatisfying condition (BDna), and hence such that L1(G, ω) is a symmetric∗-algebra that admits functional calculus. We shall first prove that in this situation the setm(∅) contains functionsϕ{fs} satisfying
kϕ{fs} ∗F−Fkω→0
in L1(G, ω), for all F ∈ Cc(G). The techniques of the proof will be the same as those used in ([6]) for the Wiener property, adapted to the new method of functional calculus.
7.3. Let (fs)sbe a bounded approximate identity inL1(G, ω) such that, for alls, fs=fs∗, kfskω≤C, kfsk1= 1, suppfs⊂Vs⊂K,
whereC is a positive constant,Vsa compact symmetric neighbourhood ofeinGand K a fixed compact set. We shall show that there exists a periodic functionϕ∈Φ of period 2πwithϕ(1) = 1,ϕ≡0 in a neighbourhood of 0, such that
ϕ{fs}=X
n∈Z
ˆ
ϕ(n)u(nfs) is converging for allsand such that
kϕ{fs} ∗F−Fkω→0
for all continuous functionsF with compact support in G. Moreover the functions ϕ{fs}are contained inm(∅) by construction.
In fact, let’s choose ϕ∈Φ such thatϕ(1) = 1. Then kϕ{fs} ∗F−Fkω=kX
n∈Z
ˆ
ϕ(n)[einfs∗F−einF]kω.
As the functions fs are uniformly bounded in L1(G, ω), it is easy to check that for every fixedn,
einfs∗F →einF
inL1(G, ω) ass→ ∞. So, for any fixedN ∈N, we have X
|n|≤N
ˆ
ϕ(n)[einfs∗F−einF]→0
in L1(G, ω). Next we have to show that we may chooseϕsuch that, for anyε >0, there existsN ∈N such that
X
|n|>N
ˆ
ϕ(n)[einfs∗F−einF]
ω
≤ X
|n|>N
|ϕ(n)|keˆ infs∗Fkω+
X
|n|>N
ˆ ϕ(n)ein
kFkω< ε, independently ofs. Suppose that we had already determinedϕandN1 such that
X
|n|>N1
|ϕ(n)|keˆ infs∗Fkω< ε
2, (4)
for alls, then (asP
n∈Zϕ(n)eˆ in=ϕ(1) = 1 converges) we can chooseN ≥N1 such
that
X
|n|>N
ˆ ϕ(n)ein
kFkω<ε 2. Thus it suffices to show (4). According to 3.12
keinfs∗Fkω≤K2(1 +|n|)(1 +|n|2)Q2s(|n|)2 ln(ln|n|)eK3(
|n|
(ln|n|)C)
with constantsK2andK3 given by
K2= 2Q2 K12(1 +q)Q2s(q)kFk2+kFkω K3= 2e(eC
0p)kfskω,
where the constants C, K, C0 just depend on the growth of the weight and where q ∈ N∗ is such that suppF ⊂ Uq. Moreover, as the support of the functions fs
is contained in a fixed compact set, there exists p ∈ N∗ such that suppfs ⊂ Up and hence kfskω ≤supx∈Upω(x)kfsk1 = supx∈Upω(x) = s(p), for all s. Hence the constants K2 and K3 may in fact be chosen independently ofs. Moreover, up to a constant, the bound is exactly the same as the one obtained for ku(nfs)kω. Hence, if the weightω satisfies (BDna), there exist functions ϕ∈Φ satisfying the required conditions such that (4) holds. This proves the following result:
Theorem 7.4. LetGbe a compactly generated, locally compact group with polynomial growth. Let ω be a weight on Gthat satisfies condition (BDna). Then L1(G, ω)has the Wiener property.
Proof. By 7.3,j(∅) contains all ofCc(G) and hence equalsL1(G, ω), as all theϕ{fs} are in m(∅) ⊂ j(∅). Finally, if I is a closed two-sided ideal of L1(G, ω) such that h(I) =∅, thenL1(G, ω) =j(∅)⊂I by 6.3.
Examples 7.5. (i) In 4.10 (i), (iii) we have examples of weights that satisfy (BDna) and hence give a group algebra L1(G, ω) that has the Wiener property.
(ii) Examples 4.10 (iv), (v) give certain growth conditions on the coefficientsci for the corresponding weightωto satisfy (BDna). Under these conditionsL1(G, ω) has the Wiener property.
(iii) Letω(x) =eC
|x|
(ln(|x|+1))γ. Thenω satisfies (BDna) if and only if γ >1. Hence, for γ >1, L1(G, ω) has the Wiener property. This is the best possible result.
In fact, if G=R, the result of Vretblad ([20]) shows that ifL1(R, ω) has the Wiener property, thenγ >1. So even ifG=R, we get the same condition on γ that we had already in this paper for the non abelian case.
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