Functional calculus in weighted group algebras
Functional calculus in weighted group algebras
Functional calculus in weighted group algebras
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For the <strong>group</strong> G = R, Vretblad even proves a partial converse of the Wiener property. Heshows the follow<strong>in</strong>g: 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 decreas<strong>in</strong>g on R + and ω <strong>in</strong>creas<strong>in</strong>gon R + . Then L 1 (R, ω) does not have the Wiener property. Let’s write nx <strong>in</strong>stead of x n ,as the <strong>group</strong> 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 <strong>in</strong> prov<strong>in</strong>g similar results for nonabelian <strong>group</strong>s. Let G be a locally compact <strong>group</strong> 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 follow<strong>in</strong>g: Let N be an arbitrary open set <strong>in</strong> Ĝ. Doesthere exist f N <strong>in</strong> the <strong>weighted</strong> <strong>group</strong> 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 <strong>in</strong> 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 <strong>group</strong> G and on the growth of the weight ω.Before stat<strong>in</strong>g our results, let us recall some precise def<strong>in</strong>itions. A weight ω on a locallycompact <strong>group</strong> 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 <strong>in</strong> 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 <strong>in</strong> ([Fe.Gr.Lei.Lu.Mo.]), noth<strong>in</strong>g seems to be known for non-abelian <strong>group</strong>s and for weights grow<strong>in</strong>gfaster than sub-exponential ones. One may ask the question of what are the most generalweights ω on a locally compact <strong>group</strong> G, such that the <strong>weighted</strong> <strong>group</strong> 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 <strong>group</strong>swith polynomial growth. For such a <strong>group</strong>, let U be a generat<strong>in</strong>g 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 def<strong>in</strong>e the follow<strong>in</strong>g quantities:where N ∗ = N \ {0} (see 1.2.) andτ U (x) = |x| = <strong>in</strong>f{n ∈ N ∗ | x ∈ U n }s(n) = supx∈U n ω(x).If G = R, τ U (·) = | · | is equivalent to the absolute value <strong>in</strong> the follow<strong>in</strong>g 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 <strong>calculus</strong> to constructan approximate identity conta<strong>in</strong>ed <strong>in</strong> the m<strong>in</strong>imal 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 grow<strong>in</strong>g 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 m<strong>in</strong>imal 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 grow<strong>in</strong>g weights, thepresence of the factor (ln(ln n)) does not seem to affect the result as the follow<strong>in</strong>g exampleshows: Let ω be def<strong>in</strong>ed 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 grow<strong>in</strong>g weights ourresult seems to be almost optimal, as we get the same condition on γ as <strong>in</strong> the abeliancase. Another weight studied <strong>in</strong> 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 obta<strong>in</strong>ed thanks to the construction of a functional <strong>calculus</strong> on L 1 (G, ω). Ingeneral, functional (or symbolic) <strong>calculus</strong> <strong>in</strong> function <strong>algebras</strong> is a special case of functional<strong>calculus</strong> <strong>in</strong> Banach ∗-<strong>algebras</strong>. Let’s recall the follow<strong>in</strong>g def<strong>in</strong>ition 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, conta<strong>in</strong><strong>in</strong>g 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 <strong>in</strong>stance [Hu.1], [Katz.]). We then writeg = ϕ{f}. The idea to realize this comes from the Fourier <strong>in</strong>version theorem. Let f be aself-adjo<strong>in</strong>t element of B (if B is a symmetric ∗-algebra) and let’s def<strong>in</strong>eu(nf) =∞∑ i k n kf k , n ∈ Z.k!k=1Here f k is the k-th power of f for the multiplication <strong>in</strong> the algebra B. In case of L 1 (G) orL 1 (G, ω) we write f ∗k to <strong>in</strong>dicate 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 <strong>in</strong> the Banach algebra B (‖ · ‖ = ‖ · ‖ ω <strong>in</strong> caseof L 1 (G, ω)) and where ˆϕ is the Fourier transform of ϕ on the <strong>in</strong>terval [0, 2π]. Furtherdetails about the functional <strong>calculus</strong> for function <strong>algebras</strong> and its properties will be given<strong>in</strong> 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 (<strong>in</strong> case of a <strong>weighted</strong> <strong>group</strong> algebra). In fact, <strong>in</strong>order 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 (<strong>in</strong> order to prove the desired harmonicanalysis properties).In ([Fe.Gr.Lei.Lu.Mo.]) such a functional <strong>calculus</strong> was constructed for L 1 (G, ω) if theweight ω is sub-exponential. This proof cannot be extended to faster grow<strong>in</strong>g weights.So a new approach to this problem will have to be developed <strong>in</strong> this paper. First theevaluation of the bound for ‖u(nf)‖ ω (f = f ∗ cont<strong>in</strong>uous with compact support) has to beimproved considerably. This will be done <strong>in</strong> section 3 by adapt<strong>in</strong>g and develop<strong>in</strong>g a method<strong>in</strong>troduced by Hulanicki <strong>in</strong> ([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). F<strong>in</strong>ally, there exist enough such functions ϕ to prove the harmonic analysisresults we are <strong>in</strong>terested <strong>in</strong> (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 <strong>in</strong> ([Fe.Gr.Lei.Lu.Mo.]), but theirpresence <strong>in</strong> this paper is necessary for a better understand<strong>in</strong>g of the abstract results thatwill follow.1.2. Let G be a compactly generated, locally compact <strong>group</strong> 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 generat<strong>in</strong>g 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 def<strong>in</strong>ed byτ U (x) = <strong>in</strong>f{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, +∞[ def<strong>in</strong>ed 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 def<strong>in</strong>ition is <strong>in</strong>dependent of the choice of the neighbourhood U. If G is a connected,simply connected, nilpotent Lie <strong>group</strong>, this is equivalent to the fact that ω(x) is boundedby a polynomial function <strong>in</strong> the coord<strong>in</strong>ates of the first or second k<strong>in</strong>d 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 def<strong>in</strong>ition is <strong>in</strong>dependent 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 def<strong>in</strong>ed <strong>in</strong> 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 Beurl<strong>in</strong>g ([Beu.],[Beu.1]) and Vretblad ([Vr.]) called the nonanalyticcase.b) Let now G be an arbitrary compactly generated locally compact <strong>group</strong> with polynomialgrowth. Let ω be a weight on G and U a generat<strong>in</strong>g 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 aga<strong>in</strong> def<strong>in</strong>e 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 <strong>in</strong>stance already found <strong>in</strong> ([Di.]).2.3. Instead of the ”discrete” def<strong>in</strong>ition ϕ{f} = ∑ ∫n∈Zu(nf) ˆϕ(n), one may also use a”cont<strong>in</strong>uous” def<strong>in</strong>ition 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 <strong>calculus</strong> rema<strong>in</strong> 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 <strong>in</strong> functional <strong>calculus</strong>3.1. 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 def<strong>in</strong>e three other weights that will be useful <strong>in</strong> 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 aga<strong>in</strong> a weight (bounded on compact sets).Let’s def<strong>in</strong>e 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 <strong>in</strong>creas<strong>in</strong>g function, that will only be specified later.10
For any complex valued function f def<strong>in</strong>ed on G, let’s recall thatf ∗ (x) = ∆(x −1 )f(x −1 ) = f(x −1 ), ∀x ∈ G,as ∆ ≡ 1. This def<strong>in</strong>es an <strong>in</strong>volution <strong>in</strong> the <strong>weighted</strong> <strong>group</strong> algebra L 1 (G, ω). For the restof this section f = f ∗ will be a self-adjo<strong>in</strong>t cont<strong>in</strong>uous function with compact support.F<strong>in</strong>ally, for any subset A of G, let’s note A c = G \ A. We are now <strong>in</strong> the position tostart the computation of the bound of ‖u(nf)‖ ω for all n ∈ Z. To do this we shall adaptthe methods <strong>in</strong>troduced by ([Hu.1]) to our situation.3.2. Let us use the follow<strong>in</strong>g 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 ≤1<strong>in</strong>f 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≤<strong>in</strong>f 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 <strong>in</strong>f ∗ 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 follow<strong>in</strong>g estimations (us<strong>in</strong>g 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 <strong>in</strong>tegral 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 <strong>in</strong>f ∗ 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,bandF<strong>in</strong>ally,‖(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 <strong>in</strong> 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 obta<strong>in</strong>b 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.) rema<strong>in</strong> valid even for a = 0 with the same constantsC 1 , C 2 .3.9. Us<strong>in</strong>g (3.4.), (3.7.), and (3.8.), we get the follow<strong>in</strong>g 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 <strong>in</strong>f ∗ 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 <strong>in</strong> (3.8.). F<strong>in</strong>ally,‖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 ) <strong>in</strong>15
the constant C 1 and ‖u(f)‖ 1 by max(‖u(f)‖ 1 , ‖u(−f)‖ 1 ) <strong>in</strong> 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 <strong>in</strong>f ∗ 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 obta<strong>in</strong>ed <strong>in</strong> the follow<strong>in</strong>g 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 orig<strong>in</strong>al weight ω and on the functions fand F .3.11. Choice of the radius r(n): Motivated by the techniques of functional <strong>calculus</strong>exposed later on <strong>in</strong> this paper, let’s putr(n) = [ln(ln n) + 1], ∀n ≥ e e ,where [z] denotes the <strong>in</strong>teger 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 follow<strong>in</strong>g bounds:16
3.12. Theorem: Let G be a locally compact, compactly generated <strong>group</strong> with polynomialgrowth. Let ω be an arbitrary weight on G. Let f = f ∗ : G → C be cont<strong>in</strong>uous with compactsupport. Let F : G → C be an arbitrary cont<strong>in</strong>uous function with compact support. Wehave the follow<strong>in</strong>g 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 <strong>in</strong>f ∗ 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 generat<strong>in</strong>g 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. <strong>Functional</strong> <strong>calculus</strong>4.1. In this section we shall first construct functions ϕ whose Fourier transform havea strong decrease and may hence operate, under certa<strong>in</strong> conditions, on the self-adjo<strong>in</strong>tfunctions of Cc ∞ . This will be done us<strong>in</strong>g a result of Paley-Wiener. The use of thesefunctions <strong>in</strong> connection with the bound of ‖u(nf)‖ ω obta<strong>in</strong>ed <strong>in</strong> (3.), will give us a conditionon the growth of the weight ω. This condition will ensure the existence of functional<strong>calculus</strong> on a total part of L 1 (G, ω) and will be called the non-abelian Beurl<strong>in</strong>g-Domarcondition, because of its similarity with the results of Beurl<strong>in</strong>g and Domar.4.2. A result of Paley-Wiener: For any function f ∈ L 2 (R) the follow<strong>in</strong>g twopropositions are equivalent:(i) There is a function g ∈ L 2 (R) such that |g| = |f| and ĝ vanishes on a half l<strong>in</strong>e.(ii)∫| ln |f(x)||1 + x 2 dx < +∞See ([P.W.]) or ([Rei.]).We shall apply the previous result <strong>in</strong> the follow<strong>in</strong>g 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 translat<strong>in</strong>g 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 tak<strong>in</strong>g 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].F<strong>in</strong>ally the function h = g 1 ∗ g 2 is <strong>in</strong> 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 <strong>in</strong>verse 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 (depend<strong>in</strong>g on r), then ˆϕ 1 ∈ L 1 (R, |x r |) for all r ∈ Nand ϕ 1 ∈ Cc∞ (R). Moreover, translat<strong>in</strong>g the function ϕ 1 doesn’t change the growth of itsFourier transform, so the Fourier transform of the translated function stays <strong>in</strong> 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 <strong>in</strong>terval 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 <strong>in</strong>tervals [q, r] ⊂ [q − ε, r + ε]. It is then possible to construct a cont<strong>in</strong>uousfunction 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 f<strong>in</strong>d 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, solv<strong>in</strong>g 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 <strong>in</strong>creas<strong>in</strong>g 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 follow<strong>in</strong>g 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 <strong>in</strong>creas<strong>in</strong>g 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 follow<strong>in</strong>g 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 <strong>group</strong> 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) <strong>in</strong>Prim ∗ 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 <strong>weighted</strong> <strong>group</strong> algebra is given by the follow<strong>in</strong>g result:5.7. Theorem: Let G be a compactly generated, locally compact <strong>group</strong> with polynomialgrowth. Let ω be a weight on G. If ω satisfies condition (S), then the algebra L 1 (G, ω) issymmetric. This is <strong>in</strong> 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 follow<strong>in</strong>g theorem:5.8. Theorem: Let G be a compactly generated, locally compact <strong>group</strong> with polynomialgrowth. Let ω be a weight on G satisfy<strong>in</strong>g (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. M<strong>in</strong>imal ideals6.1. In this chapter we are go<strong>in</strong>g to study the existence of m<strong>in</strong>imal ideals of L 1 (G, ω) ofa given hull <strong>in</strong> Prim ∗ L 1 (G, ω). This study relies ma<strong>in</strong>ly on the work of ([Lu.]). Moreover30
it is connected to the problem of the Wiener property which we shall consider <strong>in</strong> chapter7. The hypotheses on the <strong>group</strong> G and the weight ω will be the same as <strong>in</strong> 5. For agiven closed subset C of Prim ∗ L 1 (G, ω) (identified with Ĝ), let’s <strong>in</strong>troduce the follow<strong>in</strong>gnotations:‖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 <strong>in</strong> ([Lu.]) gives the follow<strong>in</strong>g 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 replac<strong>in</strong>g f by f ∗ f ∗ ) and that ‖f‖ 1 ≤ 1 (bydivid<strong>in</strong>g 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 <strong>in</strong> 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 m<strong>in</strong>imal ideals of a given hull, as stated <strong>in</strong> the follow<strong>in</strong>g theorem:Theorem: Let G be a compactly generated, locally compact <strong>group</strong> 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 isconta<strong>in</strong>ed <strong>in</strong> 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 follow<strong>in</strong>g def<strong>in</strong>ition (see [Fe.Gr.Lei.Lu.Mo.]):7.1. Def<strong>in</strong>ition: 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 <strong>group</strong> G, we also say that the <strong>group</strong> 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 <strong>group</strong> 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 compact<strong>group</strong> 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 <strong>group</strong>s, 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 <strong>in</strong> particular the case if ∑ +∞ ln(s(n))2 n=1 n< +∞.2For G = R, Vretblad ([Vr.]) even shows the converse, for a certa<strong>in</strong> type of weights: Ifω(x) = exp( π 2 |x|q(x)) with q decreas<strong>in</strong>g on R + and ω <strong>in</strong>creas<strong>in</strong>g on R + and if L 1 (R, ω)has the Wiener property, then ∑ +∞ ln(ω(nx))n=1 n< +∞ (we write nx <strong>in</strong>stead of x n as G = R)2for every x ∈ R (see the <strong>in</strong>troduction for more details).In this section we shall study the Wiener property for <strong>algebras</strong> of the form L 1 (G, ω),where G is a compactly generated, locally compact <strong>group</strong> with polynomial growth and ωis a weight on G satisfy<strong>in</strong>g condition (BDna), and hence such that L 1 (G, ω) is a symmetric∗-algebra that admits functional <strong>calculus</strong>. We shall first prove that <strong>in</strong> this situation theset m(∅) conta<strong>in</strong>s functions ϕ{f s } satisfy<strong>in</strong>g‖ϕ{f s } ∗ F − F ‖ ω → 0<strong>in</strong> L 1 (G, ω), for all F ∈ C c (G). The techniques of the proof will be the same as those used<strong>in</strong> ([Fe.Gr.Lei.Lu.Mo.]) for the Wiener property, adapted to the new method of functional<strong>calculus</strong>.7.3. Let (f s ) s be a bounded approximate identity <strong>in</strong> 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 <strong>in</strong> 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 <strong>in</strong> a neighbourhood of 0, such thatϕ{f s } = ∑ n∈Zˆϕ(n)u(nf s )32
is converg<strong>in</strong>g for all s and such that‖ϕ{f s } ∗ F − F ‖ ω → 0for all cont<strong>in</strong>uous functions F with compact support <strong>in</strong> G. Moreover the functions ϕ{f s }are conta<strong>in</strong>ed <strong>in</strong> 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 <strong>in</strong>f s∗ F − e <strong>in</strong> F ]‖ ω .As the functions f s are uniformly bounded <strong>in</strong> L 1 (G, ω), it is easy to check that for everyfixed n,e <strong>in</strong>f s∗ F → e <strong>in</strong> F<strong>in</strong> L 1 (G, ω) as s → ∞. So, for any fixed N ∈ N, we have∑|n|≤Nˆϕ(n)[e <strong>in</strong>f s∗ F − e <strong>in</strong> F ] → 0<strong>in</strong> 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 <strong>in</strong>f s∗ F − e <strong>in</strong> F ]‖ ω ≤ ∑∑| ˆϕ(n)|‖e <strong>in</strong>f s∗ F ‖ ω + | ˆϕ(n)e <strong>in</strong> |‖F ‖ ω|n|>N< ε,|n|>N|n|>N<strong>in</strong>dependently of s. Suppose that we had already determ<strong>in</strong>ed ϕ and N 1 such that∑|n|>N 1| ˆϕ(n)|‖e <strong>in</strong>f s∗ F ‖ ω < ε 2 ,for all s, then (as ∑ n∈Z ˆϕ(n)e<strong>in</strong> = ϕ(1) = 1 converges) we can choose N ≥ N 1 such that|∑|n|>Nˆϕ(n)e <strong>in</strong> |‖F ‖ ω < ε 2 .Thus it suffices to show (∗). Accord<strong>in</strong>g to (3.12.)(∗)‖e <strong>in</strong>f 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 conta<strong>in</strong>ed <strong>in</strong>a 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 may<strong>in</strong> fact be chosen <strong>in</strong>dependently of s. Moreover, up to a constant, the bound is exactly thesame as the one obta<strong>in</strong>ed for ‖u(nf s )‖ ω . Hence, if the weight ω satisfies (BDna), thereexist functions ϕ ∈ Φ satisfy<strong>in</strong>g the required conditions such that (∗) holds. This provesthe follow<strong>in</strong>g result:7.4. Theorem: Let G be a compactly generated, locally compact <strong>group</strong> 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(∅) conta<strong>in</strong>s all of C c (G) and hence equals L 1 (G, ω), as all the ϕ{f s }are <strong>in</strong> m(∅) ⊂ j(∅). F<strong>in</strong>ally, 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 <strong>group</strong> algebra L 1 (G, ω) that has the Wiener property.b) Examples (4.10. d), e)) give certa<strong>in</strong> growth conditions on the coefficients c i for thecorrespond<strong>in</strong>g 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 <strong>in</strong> this paper for thenon abelian case.References[ Beu.] Beurl<strong>in</strong>g, A., Sur les <strong>in</strong>tégrales de Fourier absolument convergentes et leur applicationà une transformation fonctionnelle, 9ème congrès des mathématiciens scand<strong>in</strong>aves,août 1938, Hels<strong>in</strong>ki, 345-366, (1939).[ Beu.1] Beurl<strong>in</strong>g, A., Sur les spectres des fonctions, Colloques <strong>in</strong>ternationaux duC.N.R.S., Analyse harmonique, Nancy, 9-29, (1947).[ Boi.] Boidol, J., Lept<strong>in</strong>, H., Schürman, J., Vahle, D., Räume primitiver Ideale vonGruppenalgebren, Math. Ann., 236, 1-13, (1978).[ Di.] Dixmier, J., Opérateurs de rang f<strong>in</strong>i dans les représentations unitaires, Publ.Math. IHES 6, 305-317, (1960).[ Do.] Domar, Y., Harmonic analysis based on certa<strong>in</strong> commutative Banach <strong>algebras</strong>,Acta Mathematica, 96, 1-66, (1956).[ Fe.Gr.Lei.Lu.Mo.] Fendler, G., Gröchenig, K., Le<strong>in</strong>ert, M., Ludwig, J., Molitor-Braun,C., Weighted Group Algebras on Groups of polynomial Growth, accepted by Math.Zeitschrift.34
[ Hu.] Hulanicki, A., Sub<strong>algebras</strong> of L 1 (G) associated with the Laplacian on a Lie<strong>group</strong>, Colloq. Math. 31, 259-287, (1974).[ Hu.1] Hulanicki, A., A functional <strong>calculus</strong> for Rockland operators on nilpotent Lie<strong>group</strong>s, Studia Math. 78, 253-266, (1984).[ Ka.] Kahane, J.-P., Séries de Fourier absolument convergentes, Spr<strong>in</strong>ger, Berl<strong>in</strong>,(1970).[ Katz.] Katznelson, Y., An <strong>in</strong>troduction to harmonic analysis, J. Wiley and Sons, NewYork, (1968).[ Lu.] Ludwig, J., Polynomial growth and ideals <strong>in</strong> <strong>group</strong> <strong>algebras</strong>, manuscripta math.30, 215-221, (1980).[ Lu.1] Ludwig, J., M<strong>in</strong>imal C ∗ -Dense Ideals and Algebraically Irreducible Representationsof the Schwartz-Algebra of a Nilpotent Lie <strong>group</strong>, Harmonic Analysis, LectureNotes <strong>in</strong> Math. 1359, Spr<strong>in</strong>ger 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.] M<strong>in</strong>t Elhacen, S., Molitor-Braun, C., Etude de l’algèbre L 1 ω(G) avec G <strong>group</strong>ede 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 <strong>in</strong> the complex doma<strong>in</strong>, ColloquiumPublications 19, AMS, New York, (1934).[ Py.] Pytlik, T., On the spectral radius of elements <strong>in</strong> <strong>group</strong> <strong>algebras</strong>, Bull. Acad.Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21, 899-902, (1973).[ Py.1] Pytlik, T., Symbolic <strong>calculus</strong> on <strong>weighted</strong> <strong>group</strong> <strong>algebras</strong>, Studia Math. LXIII,169-176, (1982).[ Rei.] Reiter, H., Classical Harmonic Analysis and Locally Compact Groups, OxfordUniv. Press, (1967).[ Vr.] Vretblad, A., Spectral analysis <strong>in</strong> <strong>weighted</strong> 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.frSém<strong>in</strong>aire de mathématiqueCentre Universitaire de Luxembourg162A, Avenue de la FaïencerieL-1511 LuxembourgLuxembourgmolitor@cu.lu35