Functional calculus in weighted group algebras

For the group G = R, Vretblad even proves a partial converse of the Wiener property. Heshows the following: Let ω be a symmetric weight on R such that∫Rln ω(x)dx = ∞.1 + x2 Assume moreover that ω(x) = exp( π 2 |x|q(x)) with q decreasing on R + and ω increasingon R + . Then L 1 (R, ω) does not have the Wiener property. Let’s write nx instead of x n ,as the group R is additive. Notice that under the assumptions on q and ω,+∞∑n=1ln ω(nx)n 2 < +∞ ∀x ∈ R ⇐⇒∫Rln ω(x)dx < +∞.1 + x2 Hence Vretblad shows that if L 1 (R, ω) has the Wiener property, then ∑ +∞ ln ω(nx)n=1 n< +∞,2for every x ∈ R. The aim of the present paper consists in proving similar results for nonabelian groups. Let G be a locally compact group and Ĝ its dual, i.e. the space ofequivalence classes of unitary topologically irreducible representations of G. The questionabout Domar’s property is then the following: Let N be an arbitrary open set in Ĝ. Doesthere exist f N in the weighted group algebra L 1 (G, ω) such that π(f N ) = 0 for everyπ ∈ Ĝ \ N and ρ(f N) ≠ 0 for some given ρ ∈ N? The question about Wiener’s propertyreads: If I is a proper closed two-sided ideal in L 1 (G, ω), does there exist π ∈ Ĝ such thatI ⊂ kerπ = {f ∈ L 1 (G, ω) | π(f) = 0}? The answer to both questions depends of courseon the group G and on the growth of the weight ω.Before stating our results, let us recall some precise definitions. A weight ω on a locallycompact group G is a Borel function that satisfiessuch thatω : G −→ [1, +∞[ω(xy) ≤ ω(x)ω(y) ∀x, y ∈ G.All weights will be assumed to be symmetric in this paper, i. e.ω(x −1 ) = ω(x) ∀x ∈ G.Moreover we shall always require the function ω to be bounded on compact sets.Recall also that the algebra L 1 (G, ω) is then given by{∫}L 1 (G, ω) = f : G → C | f measurable and ‖f‖ ω = |f(x)| |ω(x)| dx < +∞ .Except for the question of the symmetry of the algebra L 1 (G, ω), studied in ([Fe.Gr.Lei.Lu.Mo.]), nothing seems to be known for non-abelian groups and for weights growingfaster than sub-exponential ones. One may ask the question of what are the most generalweights ω on a locally compact group G, such that the weighted group algebra L 1 (G, ω) hasreasonable harmonic analysis properties such as Domar’s property or Wiener’s property.2G

The results known for the constant weight ω ≡ 1 and the necessity to measure the growthof the weight urge us to limit ourselves to locally compact, compactly generated groupswith polynomial growth. For such a group, let U be a generating neighbourhood of theidentity element e, with compact closure, i.e. such thatG =and such that the Haar measure of U satisfies∞⋃n=1U n|U n | ≤ K · n Qfor some K > 0, Q ∈ N. One may define the following quantities:where N ∗ = N \ {0} (see 1.2.) andτ U (x) = |x| = inf{n ∈ N ∗ | x ∈ U n }s(n) = supx∈U n ω(x).If G = R, τ U (·) = | · | is equivalent to the absolute value in the following sense: IfU = [−1, 1], then τ U (x) − 1 < |x| ≤ τ U (x) where |x| denotes the absolute value of x. Thisjustifies the use of the notation | · |.A weight ω is said to be sub-exponential of degree at most α, 0 ≤ α < 1, if there existsC > 0 such thatω(x) ≤ e C|x|α , ∀x ∈ G(see 1.4.). The way to prove the Wiener property is to use functional calculus to constructan approximate identity contained in the minimal ideal with empty hull, whichimplies, together with the symmetry of the algebra L 1 (G, ω), the Wiener property. In([Fe.Gr.Lei.Lu.Mo.]) it is shown that if ω is sub-exponential, then L 1 (G, ω) has Wiener’sproperty. In this paper we prove the Wiener property even for faster growing weights, aswell as other harmonic analysis properties of L 1 (G, ω) (homeomorphism of Prim ∗ L 1 (G)and Prim ∗ L 1 (G, ω), Domar’s property, existence of minimal ideals of a given hull). Weshow that if the weight ω satisfies the condition∑ (ln(ln n)) · ln(s(n))n 2 < +∞, (∗)n∈N,n≥e ethen L 1 (G, ω) is symmetric and has Domar’s and Wiener’s properties. The condition (∗)seems slightly stronger than Domar’s condition. Nevertheless, for fast growing weights, thepresence of the factor (ln(ln n)) does not seem to affect the result as the following exampleshows: Let ω be defined byω(x) = e C |x|(ln(|x|+1)) γ .3

Condition (∗) is satisfied for this weight if and only if γ > 1. In this case, L 1 (G, ω) hasDomar’s and Wiener’s properties. On the other hand, if G = R and ω is as above, then theresults of Domar and Vretblad show that L 1 (R, ω) has Domar’s and Wiener’s propertiesif and only if γ > 1. This is the same condition on γ. Hence, for fast growing weights ourresult seems to be almost optimal, as we get the same condition on γ as in the abeliancase. Another weight studied in this paper is∑ω(x) = ec k|x| γ kk∈Nwhere 0 < γ k < 1, γ k ↑ 1, c k > 0, ∑ k∈N c k < +∞.Our results are obtained thanks to the construction of a functional calculus on L 1 (G, ω). Ingeneral, functional (or symbolic) calculus in function algebras is a special case of functionalcalculus in Banach ∗-algebras. Let’s recall the following definition and facts: Let B bea Banach ∗-algebra. We say that a function ϕ operates on an element f of B, if theGelfand transform of f with respect to the smallest commutative Banach subalgebra Aof B, containing f, is real and if there exists g ∈ A such that ϕ ◦ ˆf = ĝ, where ˆf and ĝdenote the Gelfand transforms of f and g (see for instance [Hu.1], [Katz.]). We then writeg = ϕ{f}. The idea to realize this comes from the Fourier inversion theorem. Let f be aself-adjoint element of B (if B is a symmetric ∗-algebra) and let’s defineu(nf) =∞∑ i k n kf k , n ∈ Z.k!k=1Here f k is the k-th power of f for the multiplication in the algebra B. In case of L 1 (G) orL 1 (G, ω) we write f ∗k to indicate that it is a convolution product. A periodic function ϕof period 2π and such that ϕ(0) = 0 then operates on f through the formulaϕ{f} = ∑ n∈Zu(nf) ˆϕ(n)provided∑‖u(nf)‖| ˆϕ(n)|n∈Zconverges, where ‖ · ‖ denotes the norm in the Banach algebra B (‖ · ‖ = ‖ · ‖ ω in caseof L 1 (G, ω)) and where ˆϕ is the Fourier transform of ϕ on the interval [0, 2π]. Furtherdetails about the functional calculus for function algebras and its properties will be givenin section 2. Let’s mention here among others the pioneer work of Dixmier (1960, [Di.]),Kahane (1970, [Ka.]), Pytlik (1973 [Py.]; 1982, [Py.1]) and Hulanicki (1974, [Hu.]; 1984,[Hu.1]).To prove the convergence of ∑‖u(nf)‖| ˆϕ(n)|n∈Zone needs of course a good bound for ‖u(nf)‖ and a function ϕ whose Fourier coefficientsdecrease rapidly enough. The existence of such functions ϕ ≢ 0 will depend on the growth4

of ‖u(nf)‖, and hence on the weight ω (in case of a weighted group algebra). In fact, inorder to have ϕ ≢ 0, the decrease of | ˆϕ(n)| must not be too rapid, especially as we alsoneed some control on the support of the function ϕ (in order to prove the desired harmonicanalysis properties).In ([Fe.Gr.Lei.Lu.Mo.]) such a functional calculus was constructed for L 1 (G, ω) if theweight ω is sub-exponential. This proof cannot be extended to faster growing weights.So a new approach to this problem will have to be developed in this paper. First theevaluation of the bound for ‖u(nf)‖ ω (f = f ∗ continuous with compact support) has to beimproved considerably. This will be done in section 3 by adapting and developing a methodintroduced by Hulanicki in ([Hu.1]). Secondly, appropriate functions ϕ that operate on fhave to be constructed. To do this we shall use a result of Paley-Wiener ([P.W.]) and showthat such functions exist if the growth of the weight ω is limited by the condition∑ (ln(ln n)) ln(s(n))1 + n 2 < +∞,n∈N,n≥e e(see section 4). Finally, there exist enough such functions ϕ to prove the harmonic analysisresults we are interested in (sections 5,6,7).1. Examples of weights1.1. In this chapter we recall some results of ([Fe.Gr.Lei.Lu.Mo.]) and give some examplesof weights. Part of these examples are already found in ([Fe.Gr.Lei.Lu.Mo.]), but theirpresence in this paper is necessary for a better understanding of the abstract results thatwill follow.1.2. Let G be a compactly generated, locally compact group with polynomial growth. LetU be an open, symmetric and relatively compact neighbourhood of the identity element eof G with G = ⋃ +∞n=1 U n . Such a neighbourhood will be called a generating neighbourhood.As G has polynomial growth, there exist constants Q ∈ N, and K > 0 such that the Haarmeasure of U n satisfies|U n | ≤ Kn Q .Let τ U : G → [1, +∞[ be defined byτ U (x) = inf{n ∈ N ∗ | x ∈ U n }.Thenτ U (x · y) ≤ τ U (x) + τ U (y).It is easy to check that ω U : G → [1, +∞[ defined byω U (x) = 1 + τ U (x)5

is a weight on G. The same is true for every ω U (x) α , α ∈ R + (see [Hu.], [Hu.1], [Lu.1]).1.3. Polynomial weights: A weight ω : G → [1, +∞[ is said to be polynomial if andonly if there exists α > 0 and C > 0 such thatω(x) ≤ C(1 + τ U (x)) α , ∀x ∈ G.This definition is independent of the choice of the neighbourhood U. If G is a connected,simply connected, nilpotent Lie group, this is equivalent to the fact that ω(x) is boundedby a polynomial function in the coordinates of the first or second kind of x.1.4. Sub-exponential weights: A weight ω : G → [1, +∞[ is said to be sub-exponentialof degree at most α, 0 ≤ α < 1, if there exists C > 0 such thatω(x) ≤ e C(τ U (x)) α , ∀x ∈ G.This definition is independent of the choice of the neighbourhood U. Polynomial weightsare sub-exponential and the function0 ≤ α < 1, itself is a sub-exponential weight.x ↦→ e C(τ U (x)) α ,1.5. Condition (S): a) The condition (S) on the weight ω is defined in order to get thesymmetry of the algebra L 1 (G, ω). A clue is given by what happens for L 1 (R, ω), whereω(x) = e Φ(|x|) with Φ : R + → R + . One knows that L 1 (R, ω) is symmetric if and only ifL 1 (R, ω) admits no non-unitary characters (i.e. no characters x ↦→ e cx where c ∈ C has anon-zero real part), which is equivalent to the fact thatln ω(x)lim|x|→+∞ |x|Φ(|x|)= lim|x|→+∞ |x|This condition is what Beurling ([Beu.],[Beu.1]) and Vretblad ([Vr.]) called the nonanalyticcase.b) Let now G be an arbitrary compactly generated locally compact group with polynomialgrowth. Let ω be a weight on G and U a generating neighbourhood. If the weight ω isradial for U, i.e. if ω(x) = ω(τ U (x)) = e Φ(τ U (x)) for some function Φ : N → R + , then wemay again define condition (S) byΦ(τ U (x))limτ U (x)→+∞ τ U (x)Φ(k)= limk→+∞ k= 0.= 0.If the weight is not radial, we have to generalize the previous condition as follows.6

To see this, one may apply the left regular representation, which is simple, to (ϕ · ψ){f}.In particular, if ϕ is such that suppϕ ∩ [0, 2π] ⊂]0, 2π[ and such that ˆϕ has an appropriatedecrease, then, there exists a function ψ with the same properties and such that ψ ≡ 1 onsuppϕ. Henceϕ{f} ∗ ψ{f} = ψ{f} ∗ ϕ{f} = (ϕ · ψ){f} = ϕ{f}.Similar results are for instance already found in ([Di.]).2.3. Instead of the ”discrete” definition ϕ{f} = ∑ ∫n∈Zu(nf) ˆϕ(n), one may also use a”continuous” definition given by ϕ{f} = 12π R u(λf) ˆϕ(λ)dλ for a real-valued C∞ -functionϕ with compact support, such that ϕ(0) = 0 and such that ˆϕ has an appropriate decrease.The properties of functional calculus remain of course the same. Moreover, one may alsouse results of Mandelbrojt ([Ma.], [Ma.1]) to construct such functions ϕ.3. Computation of the bound used in functional calculus3.1. Let ω be an arbitrary weight on G (bounded on compact sets). Then there exists aconstant C > 0 such thatω(x) ≤ e C|x| , ∀x ∈ G,because every weight is exponentially bounded. For further purposes, let’s fix this constantC > 1, which is always possible. We define three other weights that will be useful in thecomputation of the bound. First, let’s putω 1 (x) = s(|x|) =sup ω(y),y∈U |x|ω 2 (x) = e C|x| , ∀x ∈ G.Let’s also choose a constant C ′ > 0 such that(1 + |x|) Q 2 ω1 (x)ω 2 (x) ≤ e C′ |x| .This is possible because the left hand side is again a weight (bounded on compact sets).Let’s define the weightω 3 (x) = e C′ |x| , ∀x ∈ Gand let’s putLets 2 (n) = supx∈U n ω 2 (x) = e Cn ,s 3 (n) = supx∈U n ω 3 (x) = e C′n , ∀n ∈ N ∗ .r : N → Nbe an arbitrary increasing function, that will only be specified later.10

For any complex valued function f defined on G, let’s recall thatf ∗ (x) = ∆(x −1 )f(x −1 ) = f(x −1 ), ∀x ∈ G,as ∆ ≡ 1. This defines an involution in the weighted group algebra L 1 (G, ω). For the restof this section f = f ∗ will be a self-adjoint continuous function with compact support.Finally, for any subset A of G, let’s note A c = G \ A. We are now in the position tostart the computation of the bound of ‖u(nf)‖ ω for all n ∈ Z. To do this we shall adaptthe methods introduced by ([Hu.1]) to our situation.3.2. Let us use the following decomposition of u(f):u(f) = g + kwith g = u(f) · χ Ur(n), k = u(f) · χ (U r(n) ) c, where χ A stands for the characteristic functionof the set A. For further purposes we compute‖k‖ 1 ≤1inf x∈(U r(n) ) c ω 2(x) ‖u(f)‖ ω 2≤1s 2 (r(n)) ‖u(f)‖ ω 2≤1s 2 (r(n)) ‖u(f)‖ ω 3.Similarly,∫‖k‖ ω ≤ |k(s)|ω(s)dsG∫≤ |k(s)|(1 + |s|) Q 2 ω1 (s)dsG∫= |k(s)|(1 + |s|) Q 2 ω1 (s)ds(U r(n) ) c ∫1≤inf x∈(U r(n) ) ω |k(s)|(1 + |s|) Q 2 ω1 (s)ω 2 (s)ds2(x)c (U r(n) ) c1≤s 2 (r(n)) ‖u(f)‖ ω 3.3.3. Consider the unitary map (u(f) + δ e ) : L 2 (G) −→ L 2 (G)F ↦−→ (u(f) + δ e ) ∗ F = e if ∗ F = u(f) ∗ F + F,where δ e is the Dirac measure. Obviously ‖(u(nf) + δ e ) ∗ F ‖ 2 = ‖e inf ∗ F ‖ 2 ≤ ‖F ‖ 2 forn ∈ Z. By (3.2.)‖(g + δ e ) ∗ F ‖ 2 = ‖(u(f) + δ e − k) ∗ F ‖ 2≤ (1 + ‖k‖ 1 )‖F ‖ 2(≤ 1 + ‖u(f)‖ )ω 3‖F ‖ 2s 2 (r(n))11

3.4. For n ∈ N, let us make the following estimations (using the methods of [Hu.1]):‖(δ e + u(f)) ∗n ‖ ω = ‖ ( (δ e + g) + k ) ∗n‖ωn∑ ∑≤ ‖(δ e + g) a 1∗ k b 1∗ · · · ∗ (δ e + g) a n∗ k b n‖ ω ,m=0 a,bwhere a = (a 1 , . . . , a n ) ∈ {0, 1} n , b = (b 1 , . . . , b n ) ∈ {0, 1} n , |a| = m, |b| = n − m,a i + b i = 1, (δ e + g) a j= δ e if a j = 0 and k b j= δ e if b j = 0. Fix a and b as above. Assumethat a ≠ 0. Then‖(δ e + g) a 1∗ k b 1∗ · · · ∗ (δ e + g) a n∗ k b n‖ ω∫ ∫≤ · · · ‖(δ e + g) a 1∗ δ b 1s 1∗ · · · ∗ (δ e + g) a n∗ δ b nsn‖ ωGG· |k(s 1 ) b1 · · · k(s n ) b n|ds 1 · · · ds n ,where the convention is such that there is no integral with respect to ds l if b l = 0. Onemay check that‖(δ e + g) a 1∗ δ b 1s 1∗ · · · ∗ (δ e + g) a n∗ δ b nsn‖ ω≤ ∑ ‖(δ e + g) a 1∗ δ b 1s 1∗ · · · ∗ (δ e + g) a j−1∗ δ b j−1s j−1∗ g ∗ δ b js j∗ · · · ∗ δ b nsn‖ ω + ∏a j =1Similarly,b i =1ω(s i ).‖e inf ∗ F ‖ ω = ‖(δ e + u(f)) ∗n ∗ F ‖ ωn∑ ∑≤ ‖(δ e + g) a 1∗ k b 1∗ · · · ∗ (δ e + g) a n∗ k b n∗ F ‖ ωm=0 a,bandFinally,‖(δ e + g) a 1∗ k b 1∗ · · · ∗ (δ e + g) a n∗ k b n∗ F ‖ ω∫ ∫≤ · · · ‖(δ e + g) a 1∗ δ b 1s 1∗ · · · ∗ (δ e + g) a n∗ δ b nsn∗ F ‖ ωGG· |k(s 1 ) b1 · · · k(s n ) b n|ds 1 · · · ds n .‖(δ e + g) a 1∗ δ b 1s 1∗ · · · ∗ (δ e + g) a n∗ δ b nsn∗ F ‖ ω≤ ∑ a j =1‖(δ e + g) a 1∗ δ b 1s 1∗ · · · ∗ (δ e + g) a j−1∗ δ b j−1s j−1∗ g ∗ δ b js j∗ · · · ∗ δ b nsn∗ F ‖ ω+ ∏ω(s i )‖F ‖ ω .b i =112

3.5. Let us denote B(n) = U n . Then, for a ≠ 0, the function(δ e + g) a 1∗ δ b 1s 1∗ · · · ∗ (δ e + g) a j−1∗ δ b j−1s j−1∗ g ∗ δ b js j∗ · · · ∗ δ b nsnis supported by B(nr(n) + |s 1 | b 1+ · · · + |s n | b n) with the convention that |s l | b l= 0 if b l = 0.Moreover, by (1.2.),∣ B(nr(n) + |s1 | b 1+ · · · + |s n | b n) ∣ ∏≤ K · (1 + nr(n))Q(1 + |s i |) Q .b i =1Similarly, the function (δ e + g) a 1∗ δ b 1s 1∗ · · · ∗ (δ e + g) a j−1∗ δ b j−1s j−1∗ g ∗ δ b js jsupported by B(q + nr(n) + |s 1 | b 1+ · · · + |s n | b n), if suppF ⊂ U q , and∣ B(q + nr(n) + |s1 | b 1+ · · · + |s n | b n) ∣ ≤ K · (1 + q) Q (1 + nr(n)) ∏ Q3.6. If h is any function with compact support in B(r), then∫‖h‖ ω = |h(x)|ω(x)dx ≤ s(r)‖h‖ 2 |B(r)| 1 2 .U rObviously ‖g‖ 2 ≤ ‖u(f)‖ 2 . Hence, by (3.3.)-(3.5.), we obtainb i =1∗ · · · ∗ δ b nsn∗ F is(1 + |s i |) Q .‖(δ e + g) a 1∗ δ b 1s 1∗ · · · ∗ (δ e + g) a n∗ δ b nsn‖ ω≤ ∑ (1 + ‖u(f)‖ ) a1 +···aω j−1 3· ‖u(f)‖2 · ∣∣ B(nr(n) + |s1 | b 1+ · · · + |s n | b n) ∣ 1 2sa j =1 2 (r(n))· s(nr(n) + |s 1 | b 1+ · · · + |s n | b n) + ∏ω(s i )b i =1∑ (≤ K 1 2 ‖u(f)‖2 1 + ‖u(f)‖ ) a1 +···+aω 3j−1( ) Q1 + nr(n)2sa j =1 2 (r(n))· s(nr(n) + |s 1 | b 1+ · · · + |s n | b n) ∏∏(1 + |s i |) Q 2 +b i =1b i =1ω(s i )(≤ K 1 2 ‖u(f)‖2 · n(1 + nr(n)) Q 2 · s(nr(n)) · 1 + ‖u(f)‖ ) n ∏ω 3(· (1 + |si |) Q 2 · ω1 (s i ) )s 2 (r(n))b i =1+ ∏ω(s i )b i =1((≤ K 1 2 ‖u(f)‖2 · n(1 + nr(n)) Q 2 · s(nr(n)) · 1 + ‖u(f)‖ ω 3s 2 (r(n))· ∏ ((1 + |si |) Q 2 · ω1 (s i ) ) + ∏ω(s i )b i =1b i =1) s2 (r(n))) ns 2 (r(n))≤ K 1 2 ‖u(f)‖2 · n(1 + nr(n)) Q n‖u(f)‖2 · s(nr(n)) · eω3 ·(s 2 (r(n)) ) · ∏ ((1 + |si |) Q 2 · ω1 (s i ) )+ ∏ω(s i ).b i =113b i =1

Similarly, if supp F ⊂ U q ,‖(δ e + g) a 1∗ δ b 1s 1∗ · · · ∗ (δ e + g) a n∗ δ b nsn∗ F ‖ ω≤ K 1 2 ‖u(f)‖1 ‖F ‖ 2 (1 + q) Q Qn‖u(f)‖2 s(q) · n(1 + nr(n)) 2 · s(nr(n)) · eω3 ·(s 2 (r(n)) )· ∏ ((1 + |si |) Q 2 · ω1 (s i ) ) ∏+ ‖F ‖ ω ω(s i )b i =1as ‖g ∗ δ b js j∗ · · · ∗ δ b nsn∗ F ‖ 2 ≤ ‖g‖ 1 ‖F ‖ 2 ≤ ‖u(f)‖ 1 ‖F ‖ 2 . Therefore for a ≠ 0 we haveb i =1Similarly,‖(δ e + g) a 1∗ k b 1∗ · · · ∗ (δ e + g) a n∗ k b n‖ ω≤ K 1 2 ‖u(f)‖2 · n(1 + nr(n)) Q n‖u(f)‖2 · s(nr(n)) · eω3 ·(s 2 (r(n)) )· ∏)|k(s i )|(1 + |s i |) Q 2 ω1 (s i )ds i + ∏ ∫|k(s i )|ω(s i )ds ib i =1( ∫ Gb i =1≤ K 1 2 ‖u(f)‖2 · n(1 + nr(n)) Q n‖u(f)‖2 · s(nr(n)) · eω3 ·(s 2 (r(n)) )· ∏)|k(s i )|(1 + |s i |) Q 2 ω1 (s i )ds i + ( ‖u(f)‖ ω3) b1 +···b n.s 2 (r(n))b i =1( ∫ GG‖(δ e + g) a 1∗ k b 1∗ · · · ∗ (δ e + g) a n∗ k b n∗ F ‖ ω≤ K 1 2 ‖u(f)‖1 · (1 + q) Q 2 s(q)‖F ‖2 · n(1 + nr(n)) Q n‖u(f)‖2 · s(nr(n)) · eω3 ·(s 2 (r(n)) )· ∏) ∏∫|k(s i )|(1 + |s i |) Q 2 ω1 (s i )ds i + ‖F ‖ω |k(s i )|ω(s i )ds ib i =1( ∫ G≤ K 1 2 ‖u(f)‖1 · (1 + q) Q 2 s(q)‖F ‖2 · n(1 + nr(n)) Q n‖u(f)‖2 · s(nr(n)) · eω3 ·(s 2 (r(n)) )· ∏) (‖u(f)‖|k(s i )|(1 + |s i |) Q ω3) b1 +···b2 n.ω1 (s i )ds i + ‖F ‖ωs 2 (r(n))b i =1( ∫ G3.7. As∫∫|k(s i )|(1 + |s i |) Q 2 ω1 (s i )ds i = |k(s i )|(1 + |s i |) Q 2 ω1 (s i )ds iG(U r(n) ) c ∫1≤|k(s i )|(1 + |s i |) Q 2 ω1 (s i )ω 2 (s i )ds is 2 (r(n)) (U r(n) ) c1≤s 2 (r(n)) ‖u(f)‖ ω 3,we have by (3.2.) and (3.6.), for a ≠ 0,‖(δ e + g) a 1∗ k b 1∗ · · · ∗ (δ e + g) a n∗ k b n‖ ω) (≤(K 1 2 ‖u(f)‖2 · n(1 + nr(n)) Q n‖u(f)‖2 s(nr(n)) · eω3 ·(s 2 (r(n)) ) ‖u(f)‖ω3+ 1 ·s 2 (r(n))(≤ C 1 (1 + n)(1 + nr(n)) Q n‖u(f)‖2 s(nr(n)) · eω3 ·(s 2 (r(n)) ) ‖u(f)‖ω3) b1 +···+b n,·s 2 (r(n))14b i =1G) b1 +···+b n

where C 1 = (K 1 2 ‖u(f)‖ 2 + 1). Analogously, for F supported by U q , we have‖(δ e + g) a 1∗ k b 1∗ · · · ∗ (δ e + g) a n∗ k b n∗ F ‖ ω≤ C 2 (1 + n) ( 1 + nr(n) ) Q2s(nr(n)) · e ‖u(f)‖ ω 3 ·(ns 2 (r(n)) ) ·( ‖u(f)‖ω3s 2 (r(n))with C 2 = K 1 2 ‖u(f)‖ 1 · (1 + q) Q 2 s(q)‖F ‖ 2 + ‖F ‖ ω .3.8. For a = 0, we have b 1 + · · · + b n = n andSimilarly) b1 +···+b n,‖(δ e + g) a 1∗ k b 1∗ · · · ∗ (δ e + g) a n∗ k b n‖ ω = ‖k ∗n ‖ ω ≤ ‖k‖ n ω ≤ ( ‖u(f)‖ ω3) n.s 2 (r(n))‖(δ e +g) a 1∗k b 1∗· · ·∗(δ e +g) a n∗k b n∗F ‖ ω = ‖k ∗n ∗F ‖ ω ≤ ‖F ‖ ω ‖k‖ n (‖u(f)‖ ω3) n.ω ≤ ‖F ‖ ωs 2 (r(n))This shows that the bounds of (3.7.) remain valid even for a = 0 with the same constantsC 1 , C 2 .3.9. Using (3.4.), (3.7.), and (3.8.), we get the following bounds:‖(δ e + u(f)) ∗n ‖ ω≤ C 1 (1 + n)(1 + nr(n)) Q 2 s(nr(n)) · e‖u(f)‖ ω3 ·(ns 2 (r(n)) ) ·= C 1 (1 + n)(1 + nr(n)) Q 2 s(nr(n)) · e‖u(f)‖ ω3 ·(ns 2 (r(n)) ) ·≤ C 1 (1 + n)(1 + nr(n)) Q 2 s(nr(n)) · e2‖u(f)‖ ω3 ·(Likely,ns 2 (r(n)) ) .n∑ ∑1 ·m=0 a,b(1 + ‖u(f)‖ ω 3s 2 (r(n))( ‖u(f)‖ω3) b1 +···+b ns 2 (r(n))) n‖(δ e + u(f)) ∗n ∗ F ‖ ω = ‖e inf ∗ F ‖ ω≤ C 2 (1 + n)(1 + nr(n)) Q 2 s(nr(n)) · e2‖u(f)‖ ω3 ·(The constants C 1 , C 2 are the same as in (3.8.). Finally,‖u(nf)‖ ω ≤ ‖(δ e + u(f)) ∗n ‖ ω + 1≤ 2C 1 (1 + n)(1 + nr(n)) Q 2 s(nr(n)) · e2‖u(f)‖ ω3 ·(ns 2 (r(n)) ) .ns 2 (r(n)) ) .3.10. The previous bounds have been proven for n ∈ N. If we want a bound valid for n ∈ Z,we first put nf = (−n)(−f) if n < 0. We replace ‖u(f)‖ 2 by max(‖u(f)‖ 2 , ‖u(−f)‖ 2 ) in15

the constant C 1 and ‖u(f)‖ 1 by max(‖u(f)‖ 1 , ‖u(−f)‖ 1 ) in the constant C 2 . We thenhaveand‖u(nf)‖ ω ≤ C 1(1 ′ + |n|)(1 + |n|r(|n|)) Q C2 s(|n|r(|n|)) · e ′ 3 ( s 2 (r(|n|)) ) , n ∈ Z,‖e inf ∗ F ‖ ω ≤ C 2(1 ′ + |n|)(1 + |n|r(|n|)) Q C2 s(|n|r(|n|)) · e ′ 3 ( s 2 (r(|n|)) ) , n ∈ Z,where the constants C ′ 1, C ′ 2, C ′ 3 are obtained in the following way:C ′ 1 = 2K 1 2 e‖f‖ 2+ 1≥ 2K 1 2 max(‖u(f)‖2 , ‖u(−f)‖ 2 ) + 1C ′ 2 = K 1 2 e‖f‖ 1(1 + q) Q 2 s(q)‖F ‖2 + ‖F ‖ ω≥ K 1 2 max(‖u(f)‖1 , ‖u(−f)‖ 1 )(1 + q) Q 2 s(q)‖F ‖2 + ‖F ‖ ω ,where q ∈ N ∗ is such that suppF ⊂ U q . As f has compact support, there exists p ∈ N ∗such that suppf ⊂ U p . Hence|n||n|‖f‖ ω3≤ e C′p ‖f‖ 1 ≤ e C′p ‖f‖ ωandC ′ 3 = 2e (eC′p )‖f‖ ω≥ 2e ‖f‖ ω 3 ≥ 2 max(‖u(f)‖ω3 , ‖u(−f)‖ ω3 ).So the constants depend essentially only on the original weight ω and on the functions fand F .3.11. Choice of the radius r(n): Motivated by the techniques of functional calculusexposed later on in this paper, let’s putr(n) = [ln(ln n) + 1], ∀n ≥ e e ,where [z] denotes the integer part of z. Thenln(ln |n|) ≤ r(|n|) ≤ ln(ln |n|) + 1 ≤ 2 ln(ln |n|),for |n| ≥ e eandHences 2 (r(|n|)) ≥ e C(ln(ln |n|)) = (ln |n|) C .e C′ 3|n|s 2 (r(|n|))≤ e C′ 3|n|(ln |n|) Cands(|n|r(|n|)) ≤ s(|n|) r(|n|) ≤ s(|n|) 2 ln(ln |n|) , for |n| ≥ e e .With this choice of the radius we get the following bounds:16

3.12. Theorem: Let G be a locally compact, compactly generated group with polynomialgrowth. Let ω be an arbitrary weight on G. Let f = f ∗ : G → C be continuous with compactsupport. Let F : G → C be an arbitrary continuous function with compact support. Wehave the following bounds: There exist strictly positive constants K 1 , K 2 , K 3 such that‖u(nf)‖ ω ≤ K 1 (1 + |n|)(1 + |n| 2 ) Q 2 s(|n|)2 ln(ln |n|) e K 3(‖e inf ∗ F ‖ ω ≤ K 2 (1 + |n|)(1 + |n| 2 ) Q 2 s(|n|)2 ln(ln |n|) e K 3(|n|(ln |n|) C )|n|(ln |n|) C ) ,for |n| ≥ e e . If U is a generating neighbourhood of G and p, q ∈ N ∗ such that suppf ⊂ U pand suppF ⊂ U q , thenK 1 = 2 Q 2 +1 K 1 2 e‖f‖ 2+ 1K 2 = 2 Q 2 K12 e‖f‖ 1(1 + q) Q 2 s(q)‖F ‖2 + ‖F ‖ ωK 3 = 2e (eC′p )‖f‖ ω,where the constants C, K and C ′ just depend on the growth of the weight ω and whereC > 1.Proof: By (3.10.) and (3.11.).4. Functional calculus4.1. In this section we shall first construct functions ϕ whose Fourier transform havea strong decrease and may hence operate, under certain conditions, on the self-adjointfunctions of Cc ∞ . This will be done using a result of Paley-Wiener. The use of thesefunctions in connection with the bound of ‖u(nf)‖ ω obtained in (3.), will give us a conditionon the growth of the weight ω. This condition will ensure the existence of functionalcalculus on a total part of L 1 (G, ω) and will be called the non-abelian Beurling-Domarcondition, because of its similarity with the results of Beurling and Domar.4.2. A result of Paley-Wiener: For any function f ∈ L 2 (R) the following twopropositions are equivalent:(i) There is a function g ∈ L 2 (R) such that |g| = |f| and ĝ vanishes on a half line.(ii)∫| ln |f(x)||1 + x 2 dx < +∞See ([P.W.]) or ([Rei.]).We shall apply the previous result in the following situation:R17

4.3.a) Let t : R → R be such thatt ≥ 1t(−x) = t(x)∀x ∈ Rt(x + y) ≤ Ct k (x)t k (y)∫ln(t(x))dx < +∞.1 + x2 Rfor some k ≥ 1, C > 0, ∀x, y ∈ RIf moreover, there exists l ∈ N such thatt −l ∈ L 2 (R)t −(l−k) ∈ L 1 (R),then there is a function g ∈ L 1 (R, t k ) such that suppĝ ∈ [0, +∞[.In fact, this results from Paley-Wiener if we take f = t −l .b) By translating the function ĝ we may construct a function g 1 such that |g 1 | = |g| = |f|and suppĝ 1 ⊂ [−ε, +∞[ for any given ε > 0.By taking g 2 = ǧ 1 , where ǧ 1 (x) = g 1 (−x), we get a function g 2 ∈ L 1 (R, t k ) such thatsuppĝ 2 ⊂] − ∞, ε].Finally the function h = g 1 ∗ g 2 is in L 1 (R, t) as g 1 , g 2 ∈ L 1 (R, t k ) and as t(x + y) ≤Ct k (x)t k (y). Moreover suppĥ ⊂ [−ε, ε].c) Let now ϕ 1 be the inverse Fourier transform of h. Then ϕ 1 ∈ C c (R) such that suppϕ 1 ⊂[−ε, ε] and such that ˆϕ 1 = h ∈ L 1 (R, t). If, moreover, |x r | ≤ K r t(x) for all x ∈ R, forall r ∈ N, for some constant K r (depending on r), then ˆϕ 1 ∈ L 1 (R, |x r |) for all r ∈ Nand ϕ 1 ∈ Cc∞ (R). Moreover, translating the function ϕ 1 doesn’t change the growth of itsFourier transform, so the Fourier transform of the translated function stays in L 1 (R, t).d) Assume now that ϕ 1 ∈ C c (R) such that ˆϕ 1 ∈ L 1 (R, t) and suppϕ 1 ⊂ [c, d]. Let I = [a, b]and consider ϕ = ϕ 1 ∗χ I where χ I is the characteristic function of the interval I. It is easyto check that supp ϕ ⊂ [a+c, b+d] and that ϕ ≡ 1 on [a+d, c+b] (provided a+d < c+b).Moreover,| ˆϕ(λ)| ≤ ‖ˆχ I ‖ ∞ · | ˆϕ 1 (λ)| ≤ b − a2π| ˆϕ 1(λ)|and ˆϕ ∈ L 1 (R, t). As a matter of fact,| ˆϕ(λ)| ≤ b − a2π | ˆϕ 1(λ)|= b − a2π |g 1 ∗ g 2 (λ)|≤ b − a |g| ∗ |ǧ|(λ).2π18

Let now be two intervals [q, r] ⊂ [q − ε, r + ε]. It is then possible to construct a continuousfunction ϕ, supp ϕ ⊂ [q − ε, r + ε], ϕ ≡ 1 on [q, r] and ˆϕ(λ) ∈ L 1 (R, t). We just have tocheck that it is possible to find a, b, c, d, a < b, c < d, a + d < c + b, such that⎧a + d = q⎪⎨c + b = r⎪⎩ a + c = q − εb + d = r + ε.As a matter of fact, solving this system gives us c = k, b = r − k, d = ε + k, a = q − ε − kwith k ∈ R arbitrary. If |x r | ≤ K r t(x) for all x ∈ R, for all r ∈ N, for constants K r , thenϕ ∈ Cc∞ (R).Next we shall show that there are similar results for R/2πZ, resp. Z.4.4. Let w : Z → Z be such that(1) w ≥ 1, w increasing on N(2) w(−n) = w(n) ∀n ∈ Z(3) w(n + m) ≤ Cw k (n)w k (m) for some k ≥ 1, C ≥ 1, ∀n, m ∈ N(4)∑ ln w(n)1 + n 2 < +∞.n∈NAssume moreover that there exists l 1 ∈ N such that∑(5) w −2l1k (n) < +∞(6)n∈N∑w −(l 1−k 2) (n) < +∞.n∈NThis has the following consequences for the function w:(i) First of all one hasw(n + m) ≤ Cw k (n)w k (m) ∀n, m ∈ Z.For n, m ∈ Z − this follows from the symmetry of the function w. If n ≥ 0, m ≤ 0,n + m ≥ 0, thenw(n + m) ≤ w(n) ≤ Cw k (n) ≤ Cw k (n)w k (m)as C ≥ 1, w increasing on N and k ≥ 1. The other cases are treated similarly.(ii) As k ≤ k 2 , one also has∑w −(l1−k) (n) < +∞.n∈N(iii) If we put l = l 1 · k, then l 1 ≤ l and∑w −(l−k2) (n) < +∞.n∈N19

(1 + (n + m) 2 ) Q 2 ≤((1 + n 2 )(1 + m 2 ) + nm ) Q 2e K 3(n+mand, if n ≥ m ≥ e e ,(ln(n+m)) C ) ≤ e K 3(≤≤ 2 Q 2 (1 + n 2 ) Q 2 (1 + m 2 ) Q 2( ) k ( ) k≤ 2 Q 2 (1 + n 2 ) Q 2 (1 + m 2 ) Q 2 ∀k ≥ 1, ∀n, m ∈ N,n + m(ln(n + m)) C ≤n(ln(n)) C ) e K 3((e K 3(n(ln(n)) C )) k(n(ln n) C +m(ln(m)) C )e K 3(ln(ln(n + m)) = ln(ln(n(1 + m n )))m≤ ln(ln n + ln 2)(= ln (ln n)(1 + ln 2 )ln n )≤ ln(ln n) + ln 2Similarly if m ≥ n ≥ e e . Hence, if n ≥ m ≥ e e ,m(ln m) C ,(ln(m)) C )) k, ∀k ≥ 1, ∀n, m ≥ e,≤ ln(ln n) + ln(ln m) + ln 2()≤ 2 ln(ln n) + ln(ln m) , n, m ≥ e e .s(n + m) 2 ln(ln(n+m)) ≤ s(n) 4 ln(ln(n+m))16 ln(ln n)≤ s(n)≤ s(n) 16 ln(ln n) 16 ln(ln m)s(m)=(s(n) 2 ln(ln n)) 8(s(m) 2 ln(ln m)) 8.Similarly for m ≥ n. Hence the third condition of (4.4.) is satisfied for w and for k = 8.As w(n) ≥ C(1 + n), the fifth and sixth conditions of (4.4.) are satisfied for the functionw. The fourth condition of (4.4.) admits the following equivalent formulations:∑ ln(w(n))1 + n 2 < +∞n∈N,n≥e e⇔∑ ln C 11 + n 2 + ∑ ln(1 + n)1 + n 2 + ∑n∈N,n≥e e n∈N,n≥e e⇔+ 2Q2n∈N,n≥e e∑ (ln(ln n)) ln(s(n)) ∑1 + n 2 + K 2n∈N,n≥e e∑ (ln(ln n)) ln(s(n))1 + n 2 < +∞n∈N,n≥e e23ln(1 + n 2 )1 + n 2n∈N,n≥e en(ln n) C (1 + n 2 ) < +∞

5.6. We then have Domar’s property for the algebra L 1 (G, ω):Theorem: Let G be a locally compact, compactly generated group with polynomial growth.Let ω be a weight on G that satisfies (BDna). Then, given any ρ ∈ Ĝ and any openneighbourhood N of ρ, resp. given any open neighbourhood N 1 of kerρ ∩ L 1 (G, ω) inPrim ∗ L 1 (G, ω), there exists f ∈ L 1 (G, ω) such that ρ(f) ≠ 0 and π(f) = 0 for all π ∈ Ĝ\N,resp. for all π such that kerπ ∩ L 1 (G, ω) ∈ Prim ∗ L 1 (G, ω) \ N 1 .Proof: Put C = Ĝ \ N, C 1 = Prim ∗ L 1 (G, ω) \ N 1 and apply (5.2.).The symmetry of the weighted group algebra is given by the following result:5.7. Theorem: Let G be a compactly generated, locally compact group with polynomialgrowth. Let ω be a weight on G. If ω satisfies condition (S), then the algebra L 1 (G, ω) issymmetric. This is in particular the case if ω satisfies (BDna).Proof: See ([Fe.Gr.Lei.Lu.Mo.]) for the fact that (S) implies the symmetry of the algebra.Let’s show that (BDa), and hence (BDna), implies property (S), i.e. the symmetry of theln(s(n))algebra. In fact, as the function n ↦→ s(|n|) is a weight on Z, A = lim n n≥ 0 exists.Let’s assume that A > 0. Then∑n≥1ln(s(n))1 + n 2 = ∑ ln(s(n)) n·n 1 + n 2n≥1≥ (A − ε) ∑n(1 + n −2 )n≥N 01for some ε > 0 such that A − ε > 0 and for some N 0 ∈ N. As the last sum diverges, (BDa)cannot be satisfied.The symmetry of the algebra and the previous results imply the following theorem:5.8. Theorem: Let G be a compactly generated, locally compact group with polynomialgrowth. Let ω be a weight on G satisfying (BDna). Then the spaces PrimG, Prim ∗ L 1 (G),Prim ∗ L 1 (G, ω) are homeomorphic. The spaces PrimL 1 (G) and PrimL 1 (G, ω) are homeomorphicto subspaces of Prim ∗ L 1 (G), resp. Prim ∗ L 1 (G, ω).6. Minimal ideals6.1. In this chapter we are going to study the existence of minimal ideals of L 1 (G, ω) ofa given hull in Prim ∗ L 1 (G, ω). This study relies mainly on the work of ([Lu.]). Moreover30

it is connected to the problem of the Wiener property which we shall consider in chapter7. The hypotheses on the group G and the weight ω will be the same as in 5. For agiven closed subset C of Prim ∗ L 1 (G, ω) (identified with Ĝ), let’s introduce the followingnotations:‖f‖ C = sup ‖π(f)‖ opπ∈Cm(C) = { ϕ{f} | f = f ∗ , f ∈ C c (G), ‖f‖ 1 ≤ 1, ϕ ∈ Φ,ϕ ≡ 0 on a neighbourhood of [−‖f‖ C , ‖f‖ C ] }.Let j(C) be the closed two-sided ideal of L 1 (G, ω) generated by m(C). For C = ∅ we getm(∅) = { ϕ{f} | f = f ∗ , f ∈ C c (G), ‖f‖ 1 ≤ 1, ϕ ∈ Φ,ϕ ≡ 0 on a neighbourhood of 0 }.An argument similar to the one in ([Lu.]) gives the following result:6.2. Lemma: The hull of j(C) is C.Proof: If C = ∅, C ⊂ h(j(C)). Otherwise, take π ∈ C and ϕ{f} ∈ m(C). Then,‖π(f)‖ op ≤ ‖f‖ C and π(ϕ{f}) = ϕ(π(f)) = 0, as ϕ ≡ 0 on the spectrum of π(f). Hencem(C) ⊂ kerπ, kerπ ∈ h(j(C)) and C ⊂ h(j(C)).Conversely, let ρ ∈ Ĝ \ C. By (5.2.) there exists f ∈ C c(G) such that‖f‖ C = sup ‖π(f)‖ op < ‖ρ(f)‖ op .π∈CWe may of course assume that f = f ∗ (by replacing f by f ∗ f ∗ ) and that ‖f‖ 1 ≤ 1 (bydividing by ‖f‖ 1 ). If C = ∅, replace ‖f‖ C by 0. Hence there exists ϕ ∈ Φ such that ϕ ≡ 0on a neighbourhood of [−‖f‖ C , ‖f‖ C ] and such that ϕ(‖ρ(f)‖ op ) ≠ 0. By construction,ϕ{f} ∈ m(C) and ρ(ϕ{f}) = ϕ(ρ(f)) ≠ 0 (as ‖ρ(f)‖ op is in the spectrum of ρ(f) and asϕ(‖ρ(f)‖ op ) ≠ 0). Hence kerρ ∉ h(j(C)).6.3. Because the algebra L 1 (G, ω) is also symmetric, a result of Ludwig ([Lu.]) gives usthe existence of minimal ideals of a given hull, as stated in the following theorem:Theorem: Let G be a compactly generated, locally compact group with polynomial growth.Let ω be a weight on G that satisfies condition (BDna). Let C be a closed subset ofĜ. There exists a closed two-sided ideal j(C) of L 1 (G, ω), with h(j(C)) = C, which iscontained in every two-sided closed ideal I with h(I) ⊂ C.Proof: Take ϕ{f} ∈ m(C) arbitrary. By (2.2.) and (4.8.) there exists ψ ∈ Φ suchthat ϕ · ψ = ψ · ϕ = ϕ. Hence ψ{f} ∗ ϕ{f} = ϕ{f} and ψ{f} ∈ m(C). Moreoverh({ψ{f}}) ⊃ h(m(C)) = C. We then apply lemma 2 of ([Lu.]) to conclude.31

7. Wiener propertyLet us recall the following definition (see [Fe.Gr.Lei.Lu.Mo.]):7.1. Definition: Let A be a Banach ∗-algebra. We say that A has the Wiener property(W ) if for every proper closed two-sided ideal I of A, there exists a topologically irreducible∗-representation π of A such that I ⊂ kerπ. If A is of the form L 1 (G) for some locallycompact group G, we also say that the group G has the Wiener property.7.2. Examples: a) In ([Mi.Mo.]) it is shown that if G is a connected, simply connected,nilpotent Lie group and if ω is a polynomial weight on G, then L 1 (G, ω) has the Wienerproperty.b) In ([Fe.Gr.Lei.Lu.Mo.]) it is shown that if G is a compactly generated, locally compactgroup with polynomial growth and if ω is a weight on G that is at most sub-exponential,then the algebra L 1 (G, ω) has the Wiener property.c) For abelian groups, Domar ([Do.]) has shown that L 1 (G, ω) has the Wiener property if∑ +∞ ln(ω(x n ))n=1 n< +∞ for all x ∈ G. This is in particular the case if ∑ +∞ ln(s(n))2 n=1 n< +∞.2For G = R, Vretblad ([Vr.]) even shows the converse, for a certain type of weights: Ifω(x) = exp( π 2 |x|q(x)) with q decreasing on R + and ω increasing on R + and if L 1 (R, ω)has the Wiener property, then ∑ +∞ ln(ω(nx))n=1 n< +∞ (we write nx instead of x n as G = R)2for every x ∈ R (see the introduction for more details).In this section we shall study the Wiener property for algebras of the form L 1 (G, ω),where G is a compactly generated, locally compact group with polynomial growth and ωis a weight on G satisfying condition (BDna), and hence such that L 1 (G, ω) is a symmetric∗-algebra that admits functional calculus. We shall first prove that in this situation theset m(∅) contains functions ϕ{f s } satisfying‖ϕ{f s } ∗ F − F ‖ ω → 0in L 1 (G, ω), for all F ∈ C c (G). The techniques of the proof will be the same as those usedin ([Fe.Gr.Lei.Lu.Mo.]) for the Wiener property, adapted to the new method of functionalcalculus.7.3. Let (f s ) s be a bounded approximate identity in L 1 (G, ω) such that, for all s,f s = f ∗ s , ‖f s ‖ ω ≤ C, ‖f s ‖ 1 = 1, suppf s ⊂ V s ⊂ K,where C is a positive constant, V s a compact symmetric neighbourhood of e in G and Ka fixed compact set. We shall show that there exists a periodic function ϕ ∈ Φ of period2π with ϕ(1) = 1, ϕ ≡ 0 in a neighbourhood of 0, such thatϕ{f s } = ∑ n∈Zˆϕ(n)u(nf s )32

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 )

where the constants C, K, C ′ just depend on the growth of the weight and where q ∈ N ∗is such that suppF ⊂ U q . Moreover, as the support of the functions f s is contained ina fixed compact set, there exists p ∈ N ∗ such that suppf s ⊂ U p and hence ‖f s ‖ ω ≤sup x∈U p ω(x)‖f s ‖ 1 = sup x∈U p ω(x) = s(p), for all s. Hence the constants K 2 and K 3 mayin fact be chosen independently of s. Moreover, up to a constant, the bound is exactly thesame as the one obtained for ‖u(nf s )‖ ω . Hence, if the weight ω satisfies (BDna), thereexist functions ϕ ∈ Φ satisfying the required conditions such that (∗) holds. This provesthe following result:7.4. Theorem: Let G be a compactly generated, locally compact group with polynomialgrowth. Let ω be a weight on G that satisfies condition (BDna). Then L 1 (G, ω) has theWiener property.Proof: By (7.3.), j(∅) contains all of C c (G) and hence equals L 1 (G, ω), as all the ϕ{f s }are in m(∅) ⊂ j(∅). Finally, if I is a closed two-sided ideal of L 1 (G, ω) such that h(I) = ∅,then L 1 (G, ω) = j(∅) ⊂ I by (6.3.).7.5. Examples: a) In (4.10.a), c)) we have examples of weights that satisfy (BDna) andhence give a group algebra L 1 (G, ω) that has the Wiener property.b) Examples (4.10. d), e)) give certain growth conditions on the coefficients c i for thecorresponding weight ω to satisfy (BDna). Under these conditions L 1 (G, ω) has the Wienerproperty.c) Let ω(x) = e C |x|(ln(|x|+1)) γ . Then ω satisfies (BDna) if and only if γ > 1. Hence, for γ > 1,L 1 (G, ω) has the Wiener property. This is the best possible result. In fact, if G = R, theresult of Vretblad ([Vr.]) shows that if L 1 (R, ω) has the Wiener property, then γ > 1. Soeven if G = R, we get the same condition on γ that we had already in this paper for thenon abelian case.References[ Beu.] Beurling, A., Sur les intégrales de Fourier absolument convergentes et leur applicationà une transformation fonctionnelle, 9ème congrès des mathématiciens scandinaves,août 1938, Helsinki, 345-366, (1939).[ Beu.1] Beurling, A., Sur les spectres des fonctions, Colloques internationaux duC.N.R.S., Analyse harmonique, Nancy, 9-29, (1947).[ Boi.] Boidol, J., Leptin, H., Schürman, J., Vahle, D., Räume primitiver Ideale vonGruppenalgebren, Math. Ann., 236, 1-13, (1978).[ Di.] Dixmier, J., Opérateurs de rang fini dans les représentations unitaires, Publ.Math. IHES 6, 305-317, (1960).[ Do.] Domar, Y., Harmonic analysis based on certain commutative Banach algebras,Acta Mathematica, 96, 1-66, (1956).[ Fe.Gr.Lei.Lu.Mo.] Fendler, G., Gröchenig, K., Leinert, M., Ludwig, J., Molitor-Braun,C., Weighted Group Algebras on Groups of polynomial Growth, accepted by Math.Zeitschrift.34

[ Hu.] Hulanicki, A., Subalgebras of L 1 (G) associated with the Laplacian on a Liegroup, Colloq. Math. 31, 259-287, (1974).[ Hu.1] Hulanicki, A., A functional calculus for Rockland operators on nilpotent Liegroups, Studia Math. 78, 253-266, (1984).[ Ka.] Kahane, J.-P., Séries de Fourier absolument convergentes, Springer, Berlin,(1970).[ Katz.] Katznelson, Y., An introduction to harmonic analysis, J. Wiley and Sons, NewYork, (1968).[ Lu.] Ludwig, J., Polynomial growth and ideals in group algebras, manuscripta math.30, 215-221, (1980).[ Lu.1] Ludwig, J., Minimal C ∗ -Dense Ideals and Algebraically Irreducible Representationsof the Schwartz-Algebra of a Nilpotent Lie group, Harmonic Analysis, LectureNotes in Math. 1359, Springer Verlag, 209-217, (1987).[ Ma.] Mandelbrojt, S., Séries de Fourier et classes quasi-analytiques de fonctions,Gauthier-Villars, Paris, (1935).[ Ma.1] Mandelbrojt, S., Séries adhérentes, régularisation des suites, applications,Gauthier-Villars, Paris, (1952).[ Mi.Mo.] Mint Elhacen, S., Molitor-Braun, C., Etude de l’algèbre L 1 ω(G) avec G groupede Lie nilpotent et ω poids polynomial, Publications du Centre Universitaire de Luxembourg,Travaux mathématiques X, 77-94, (1998).[ P.W.] Paley, R.E.A.C., Wiener, N., Fourier transforms in the complex domain, ColloquiumPublications 19, AMS, New York, (1934).[ Py.] Pytlik, T., On the spectral radius of elements in group algebras, Bull. Acad.Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21, 899-902, (1973).[ Py.1] Pytlik, T., Symbolic calculus on weighted group algebras, Studia Math. LXIII,169-176, (1982).[ Rei.] Reiter, H., Classical Harmonic Analysis and Locally Compact Groups, OxfordUniv. Press, (1967).[ Vr.] Vretblad, A., Spectral analysis in weighted L 1 spaces on R, Ark. Mat. 11,109-138, (1973).Institute of MathematicsDépartement de MathématiquesWroclaw UniversityUniversité de MetzPl. Grunwaldzki 2/4Ile de Saulcy50-384 Wroclaw F-57045 Metz cedex 1PolandFrancejdziuban@math.uni.wroc.plludwig@poncelet.sciences.univ-metz.frSéminaire de mathématiqueCentre Universitaire de Luxembourg162A, Avenue de la FaïencerieL-1511 LuxembourgLuxembourgmolitor@cu.lu35