C ∗ -cross products and a generalized mechanical N -body problem ∗

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C ^∗ -cross products and a generalized mechanical N -body problem ^∗

Mondher Damak & Vladimir Georgescu

Dedicated to Eyvind H. Wichmann on his seventieth birthday

Abstract

For each finite-dimensional real vector space X we construct a C^∗- algebraC₀^X graded by the lattice of all subspaces ofX. Then we compute its quotient with respect to the algebra of compact operators. This allows us to describe the essential spectrum and to prove the Mourre estimate for the self-adjoint operators associated withC₀^X.

1 Introduction

LetCbe aC^∗-algebra of bounded operators on a Hilbert spaceHand letH be a self-adjoint operator onH. One says thatH isaffiliatedtoCif (H−z)⁻¹∈C for somez ∈C\σ(H). In this case ϕ(H)∈C for allϕ∈C0(R) (the space of continuous functions which tend to zero at infinity).

Assume that the algebra of compact operators K(H) on H is included in C. Then one can construct the quotientC^∗-algebraCb=C/K(H), and one can consider the map Hb :C0(R)→Cb defined byHb(ϕ) =\ϕ(H) = [class ofϕ(H) in the quotient]. This is a morphism, andHb should be thought as an “abstract”

self-adjoint operator affiliated to the “abstract” C^∗-algebraCb. The notion of spectrum ofHb has an obvious meaning, and it is easy to show that the essential spectrum of H is given byσess(H) =σ(Hb).

This point of view is not practically useful unless the algebraCbhas a special structure. In any caseCbshould be, in some sense, simpler thanC. A systematic treatment of a rather large class of examples in which Cb and Hb are explicitly computed can be found in [4]. In particular, the relevance, in this context, of the C^∗-algebras obtained as cross products of algebras equipped with group actions is pointed out.

Our purpose in this note is to consider the quantum-mechanical N-body problem from this point of view. By taking into account the kind of potentials involved in theN-body problem it is natural (at least, a posteriori) to consider

∗Mathematics Subject Classifications: 47L65, 81Q10, 81R15, 81V70, 46L60, 46N50.

Key words: C*-cross product, N-body problem, Mourre estimate..

c2000 Southwest Texas State University and University of North Texas.

Published July 12, 2000.

51

in this case the algebraC=C₀^X constructed as follows. LetX be a real finite- dimensional vector space (the configuration space of the system ofN particles) and AtheC^∗-algebra generated by functions on X of the formϕ◦πY, where Y is a vector subspace ofX, ϕ∈C0(X/Y) and πY :X →X/Y is the natural map. The additive groupX acts continuously by translations onA. Then we takeC equal to the cross product ofAby the action ofX.

The definition of cross products used in this paper is adapted to our needs, and we do not make any explicit reference to the general theory (whose useful- ness in more complicated situations is, however, shown in [4]). LetH(X) be the space of square integrable functions onX and denote byQandP the position and momentum observables. IfAis a∗-algebra of bounded uniformly continu- ous functions onXand ifAis stable under translations, then the (norm) closed linear subspace ofB(H(X)) generated by operators of the formϕ(Q)ψ(P), with ϕ∈ A, ψ∈C0(X^∗), is aC^∗-algebra isomorphic to the “abstract” cross product AoX of A by the action of the additive group X. Here we simply take this as the definition of AoX. Thus the algebra C₀^X is obtained by choosing A equal to the closed linear space generated by functions of the formϕ◦πY with ϕ∈C0(X/Y).

In Section 2 we show that such cross products appear quite naturally in the spectral analysis of quantum Hamiltonians. Assume, for example, that one is interested in self-adjoint operators of the form H = H0+V, where H0, the kinetic energy, is an elliptic differential operator of ordermwith constant coef- ficients andV is a symmetric differential operator of order< mwith coefficients in A. Assume also that Acontains the constant functions. Then the smallest C^∗-algebra of operators onH(X) to which all these operators are affiliated (H0

being fixed) is equal to C =AoX. The main point of our approach is that this algebra often has a remarkable structure (determined by certain properties ofA), and this fact alone gives important information on the spectral proper- ties of H. Note that we constructC starting with a rather restricted class of perturbationsV. However, the class of operatorsH affiliated toC is very large and it allows quite singular perturbations (this point is studied in [2] and will not be further discussed here).

Above X^∗ is the space dual to X and one may assume without loss of generality thatX =X^∗=Rⁿ. However, our approach is explicitly independent of a choice of a Euclidean structure on the configuration space. This not only simplifies the presentation but opens the way to generalizations which allow one to study many channel Hamiltonians quite different from those ofN-body type.

Indeed, one can replaceX by a locally compact abelian group and the subspaces Y by subgroups. The case of nonabelian groups is also interesting, for example the “symplectic algebra” associated to a symplectic space (to which N-body Hamiltonians with magnetic fields are affiliated) corresponds to the Heisenberg group.

Technically speaking, the main result of this paper is Theorem 5.2. We shall state now several consequences of this theorem. The proofs are quite easy and will not be detailed (see Section 8.4.3 in [1] and note that the spectrum of a direct sum, not necessarily finite, of self-adjoint operators is equal to the closure

of the union of the spectra).

Theorem 1.1 LetH be a self-adjoint operator onH(X)affiliated toC₀^X. Then for each ω ∈ X, ω 6= 0, the limit lim_|λ|→∞τλω[H] := Hω exists in strong- resolvent sense and

σess(H) = [

ω∈X\{0}

σ(Hω). (1.1)

Here τx[H] = e^ihx,PiHe^−ihx,Pi and (e^ihx,Pif)(y) = f(x+y). Note thatHω

depends only on the one dimensional space generated by ω. In order to get a version of Theorem 1.1 which resembles more to the standard N-body version, we shall consider only a rather particular class of H. Take X =X^∗=Rⁿ and leth:Rⁿ →Rbe a continuous function such thatc⁻¹hxi^2s≤h(x)≤chxi^2s if

|x|> r, wheres, r, care strictly positive constants andhxi= (1 +|x|²)^1/2. Then H0 =h(P) is a self-adjoint operator whose form domain is the Sobolev space H^s(X). For each linear subspace Y ⊂ X let VY be a continuous symmetric sesquilinear form on H^s(X), identified with an operator H^s(X) → H^−s(X), such that:

(i) [e^ihy,Pi, VY] = 0 for ally∈Y;

(ii)k[e^ihQ,ki, VY]k_H^s_→H^−s→0 ifk∈Y,k→0 (iii)k[e^ihQ,ki−1]VYk_H^s_→H^−s →0 ifk∈Y^⊥, k→0.

We have denoted e^ihQ,kithe operator of multiplication by the function e^ihx,ki. For Y = O = {0} we take VY = 0. Note that in (ii) (but not in (iii)) the Euclidean structure of X=Rⁿ is used. Furthermore, we assume

X

Y⊂X

kVYk_H^s_→H^−s <∞,

in particular VY 6= 0 only for a countable number of subspacesY. Finally, we ask thatVY ≥ −µYH0−δY as forms onH^s(X), whereµY, δY ≥0 are numbers such thatP

Y µY <1 andP

Y δY <∞. ThenH=H0+P

Y VY is a self-adjoint, bounded from below operator onH(X),H is affiliated toC₀^X, and

σess(H) = [

Y∈M

σ(HY). Here HY = H0 +P

Z⊃Y VZ and M is the set of minimal elements (for the inclusion relation) of the class of subspaces of the formY1∩ · · · ∩Yk 6= 0 with Yi∈ {Y :VY 6= 0}.

Theorem 5.2 can also be used in order to prove the Mourre estimate for operators affiliated to C₀^X. We shall present here only the simplest case when H0is the (positive) Laplacian and theVY do not depend on the projection onY of the momentum, soVY = 1⊗V^Y ifH(X) =H(Y)⊗ H(Y^⊥). More precisely, it suffices to replace (ii) by the stronger condition:

(ii^∗) [e^ihQ,ki, VY] = 0 ifk∈Y.

Now H looks exactly as in the non-relativisticN-body problem, the only difference is that we allow an infinite lattice of subspaces. Thus we can define operators H^Y acting in H(Y^⊥) for each Y ⊂ X and these operators have a structure similar to that ofH. Note that in theN-body case the subspaces Y are usually denotedXa; moreover, one must interchange the rˆoles ofY andY^⊥ in order to agree with the conventions from [1].

Theorem 1.2 Let H be of the form described above and letD be the generator of the dilation group inH(X). Assume that the quadratic form [D,(H+i)⁻¹], with domain equal to the domain of D, extends to a bounded operator which belongs toC₀^X. LetLbe the family of subspaces of the formY1∩ · · · ∩Yk where Yi are subspaces such thatVYi6= 0. Define the set of thresholds ofH by

τ(H) = [

Y∈L\{O}

σp(H^Y).

Then the Mourre estimate forH with respect toD holds outsideτ(H).

The proof is a straightforward application of Theorem 5.2 and of Theorem 8.4.3 from [1] (infinite version). See Section 9.4.1 from [1], or [2] where the Mourre estimate is proved for more general classes of Hamiltonians.

We would like to thank George Skandalis who, several years ago, mentioned in a private discussion with one of us (VG) that the algebras defined in relation (9.2.14) from [1] are in fact cross products.

2 Cross Products and Quantum Hamiltonians

We begin with some notations and conventions adopted in this paper. If X is a locally compact topological space thenCb(X) is theC^∗-algebra of continuous bounded complex functions on it, C0(X) the C^∗-subalgebra of functions con- vergent to zero at infinity, andCc(X) the subalgebra of functions with compact support. IfHis a Hilbert space thenB(H) andK(H) are theC^∗-algebras of all bounded and compact operators onHrespectively. Bymorphism between two

∗-algebras we mean∗-morphism. IfA, B are subspaces of an algebraCthen we denote byA·B the linear subspace ofCgenerated by the elements of the form ABwithA∈A, B∈B; ifCis aC^∗-algebra then [[A·B]] is the norm closure of A·B inC. A family{Ci}i∈I of subalgebras of C islinearly independent if for each family{Si}_i∈I such thatSi∈Ci∀i, Si6= 0 for at most a finite number of iandP

i∈ISi= 0, one hasSi = 0 for alli∈I.

LetXbe a finite-dimensional real vector space. For eachx∈Xwe denote by τxthe automorphism ofCb(X) associated to the translation byx, so (τxϕ)(y) = ϕ(y+x). Then the set Cbu(X) of functions ϕ∈Cb(X) such that x7→τxϕis norm continuous is the C^∗-algebra of bounded uniformly continuous functions on X. If X = O = {0} is the zero dimensional vector space then Cb(X) = Cbu(X) =C0(X) =C.

We denote by H(X) the space of (equivalence classes of) functions on X square integrable with respect to a Haar measure (i.e. a translation invariant positive Radon measure onX). There is no canonical norm onH(X), the Haar measure being determined only modulo a positive constant factor. However, it is clear that the norms in the spaces B(H(X)) andK(H(X)) are independent of the choice of the measure. We setB(X) =B(H(X)) andK(X) =K(H(X));

these areC^∗-algebras uniquely determined byX. If X =O={0} is the zero dimensional vector space we set H(X) =CandB(O) =K(O) =C.

We embed Cb(X) in B(X) by identifying a function u ∈ Cb(X) with the operator of multiplication by uin H(X). If we denote u(Q), or u(Q^X), this operator, then the map u 7→ u(Q) is an isometric morphism of Cb(X) into B(X). So we realizeCb(X) as aC^∗-subalgebra ofB(X).

Let X^∗ be the dual space of X. For x ∈ X, x^∗ ∈ X^∗ let hx, x^∗i = x^∗(x) and identify X^∗∗ = X by setting hx^∗, xi = hx, x^∗i. We shall embed the al- gebra Cb(X^∗) of bounded continuous functions on X^∗ into B(X) by the fol- lowing procedure, explicitly independent of a Haar measure on X. It will be convenient here to use the notation e^ihx,Pi for the operator of translation by x∈ X, more precisely (e^ihx,Pif)(y) =f(x+y) (this is identical to the action of τx on functions). For eachv ∈ S(X^∗) (space of Schwartz test functions on X^∗) there is a unique measurebv on X such thatv(x^∗) =R

Xe^ihx,x∗ibv(dx). We set v(P) = R

Xe^ihx,Pibv(dx) (if X has to be specified we set v(P) = v(P^X)).

The map v 7→v(P) extends to an isometric morphism of C0(X^∗) into B(X).

This allows us to realize C0(X^∗) as a C^∗-subalgebra of B(X) and it is easy to check that the action ofC0(X^∗) onH(X) is nondegenerate. This, in turn, will give us an embedding Cb(X^∗)⊂ B(X), uniquely determined by the property u(P)v(P) = (uv)(P) ifu∈Cb, v∈C0. Observe thatC0(X^∗) is just the group C^∗-algebra of the additive groupX.

One may define the Fourier transformation as a bijective mapF :S(X)→ S(X^∗) such thatFexp(ihx, P^Xi) = exp(ihx, Q^X∗)iF for allx∈X. This defines F modulo a complex factor and Φ_X[S] =F SF⁻¹gives a canonical isomorphism Φ_X :B(X)→B(X^∗) such that Φ_X[v(P^X)] =v(Q^X∗) for allv∈Cb(X^∗).

We now prove a lemma which plays an important rˆole in our arguments.

Lemma 2.1 If u∈Cbu(X) andv ∈ S(X^∗)then for each number >0 there are points x1, . . . , xN ∈X and functionsv1, . . . , vN ∈ S(X^∗)such that

kv(P)u(Q)− XN k=1

u(Q+xk)vk(P)k< .

Proof. SetS=u(Q) andS(x) =e^ihx,PiSe^−ihx,Pi. ThenS(x) =u(Q+x),S is an element ofB(X), and the mapx7−→S(x) is bounded bykSkand is norm continuous. Choose a Haar measure dx on X. The measure bv is absolutely

continuous, so we can identify it with a function inS(X) by bv(dx)≡bv(x)dx. Then

v(P)S= Z

X

e^ihx,PiSbv(x)dx= Z

XS(x)e^ihx,Pibv(x)dx.

LetK be a compact subset ofX andV a compact neighbourhood of the origin inX. One can find functionsJ0,J1, . . . ,JN of classC^∞onX such that:

(i) 0≤Jk≤1 if 0≤k≤N andP_N

k=0Jk(x) = 1 for allx∈X; (ii)K∩supp J0=∅;

(iii) there are points x1, . . . , xN ∈ X and such that supp Jk ⊂ xk +V if 1≤k≤N.

Then we write:

v(P)S = XN k=1

S(xk) Z

X

Jk(x)e^ihx,Pivb(x)dx

+ XN k=1

Z

X

[S(x)−S(xk)]Jk(x)e^ihx,Pivb(x)dx +

Z

X

S(x)J0(x)e^ihx,Pibv(x)dx.

Let vk ∈ S(X^∗) be defined by bvk(x) =Jk(x)bv(x); then the first sum above is P_N

k=1S(xk)vk(P). Denote Kk =supp Jk and let L be the integral of|bv|. We estimate the last two terms as follows :

k XN k=1

Z

X

[S(x)−S(xk)]Jk(x)e^ihx,Pibv(x)dxk ≤ sup

1≤k≤N sup

x∈Kk

kS(x)−S(xk)kL

becauseP_N

k=1Jk(x)≤1, and k

Z

X

S(x)J0(x)e^ihx,Pibv(x)dxk ≤ kSk Z

X\K|bv(x)|dx.

If K is large enough the second member in the last inequality can be made

< /2. Finally, we have

x∈Ksupk

kS(x)−S(xk)k ≤sup

y∈VkS(y)−Sk

and this can be made< /(2L) because the functionS(·) is continuous. ♦ Corollary 2.2 If A ⊂ Cbu(X) is a translation invariant subspace, then [[A · C0(X^∗)]] = [[C0(X^∗)· A]]. In particular, if A is a translation invariant ∗- subalgebra ofCbu(X)then[[A ·C0(X^∗)]]is aC^∗-subalgebra ofB(X).

If A is a C^∗-subalgebra of Cbu(X) stable under translations then A is equipped with a continuous action of the additive groupX, so the “abstract”

C^∗-cross productAoX is well defined. It can be shown (see [4]) thatAoX is canonically isomorphic to the C^∗-subalgebra [[A ·C0(X^∗)]] ofB(X). For this reason we shall call this subalgebra cross product of A by X and we use the notation

AoX = [[A ·C0(X^∗)]]. (2.1) Our next purpose is to explain the relevance of this notion in the setting of “algebras of energy observables” considered in [4]. More precisely, we show thatAoX coincides with theC^∗-algebra generated by the Hamiltonians of the form H =H0+V whereH0, the kinetic energy, is fixed, while the interaction is given by a potentialV of “typeA” in a sense that we shall specify below. So AoX can be thought as the algebra of all Hamiltonians describing the motion of a system subject to a certain type (A) of interactions.

Let us fix aC^∗-algebra stable under translations and containing the constants A ⊂Cbu(X). We set

A^∞={ϕ∈ A ∩C^∞(X) : all the derivatives ofϕbelong toA}.

We notice that A^∞ is a dense∗-subalgebra ofA. Indeed, letπ∈Cc^∞(X) with R π(x)dx = 1 and let us set θ(x) = ⁻ⁿθ(x/) if >0. Then θ∗ϕ → ϕin sup norm as →0 ifϕ∈ A, because ϕis uniformly continuous. So it suffices to prove that all the derivatives ofθ∗ϕbelong toA. IdentifyingX =Rⁿ and taking= 1, we have forα∈Nⁿ:

(θ∗ϕ)^(α)=θ^(α)∗ϕ= Z

θ^(α)(−y)τyϕ dy.

The integral converges in norm in Cbu(X) andA is a closed subspace, so (θ∗ ϕ)^(α)∈ A.

Theorem 2.3 Let A ⊂ Cbu(X) be a translation invariant C^∗-subalgebra con- taining the constant functions and leth:X^∗→Rbe a continuous function such that |h(k)| → ∞ whenk→ ∞inX^∗. Then the C^∗-subalgebra ofB(X)gener- ated by the self-adjoint operators of the form h(P+k) +V(Q), where k ∈X^∗ andV :X→Rbelongs toA^∞, coincides withAoX.

Proof. LetCbe the C^∗-algebra generated by the operatorsH=h(P+k) + V(Q)≡H0+V(Q), withk∈X^∗ andV ∈ A^∞. By making a series expansion for largez

(z−H)⁻¹=X

n≥0

(z−H0)⁻¹[V(Q)(z−H0)⁻¹]ⁿ

we easily get C ⊂ AoX. It remains to prove the opposite inclusion. Let z ∈ C\h(X^∗). Then for µ ∈ R small enough we have z /∈ σ(Hµ) if Hµ = h(P+k)+µV(Q). The functionµ7→(Hµ−z)⁻¹is norm derivable atµ= 0 with derivative−(H0−z)⁻¹V(Q)(H0−z)⁻¹. Hence (H0−z)⁻¹V(Q)(H0−z)⁻¹∈C. Letθ∈Cc(R) with θ(0) = 1 and >0. Thenθ(H0)(H0−z) =⁻¹θ1(H0) if

θ1(t) =θ(t)(t−z/). SinceH0 is affiliated to C, we getθ(H0)(H −z)∈C. Thus:

θ(H0)V(Q)(H0−z)⁻¹=θ(H0)(H0−z)·(H0−z)⁻¹V(Q)(H0−z)⁻¹∈C.

We can write (H0−z)⁻¹ = lim_n→∞ψn(P) (norm limit) with ψn ∈ Cc(X^∗).

Then, for eachs >0,ψn(P) is a continuous map from L²(X) into the Sobolev space H^s(X) and V(Q) ∈ B(H^s(X)). Clearly lim_→0θ(H0) = 1 in norm in B(H^s(X), L²(X)). Thus we see that lim_→0θ(H0)V(Q)ψn(P) =V(Q)ψn(P) in norm inB(L²(X)) for eachn. On the other hand, we havekV(Q)ψn(P)− V(Q)(H0−z)⁻¹k →0 asn→ ∞. It follows then that lim_→0θ(H0)V(Q)(H0− z)⁻¹=V(Q)(H0−z)⁻¹ in norm inB(L²(X)).

This argument proves that

V(Q)(h(P+k)−z)⁻¹=V(Q)(H0−z)⁻¹∈C for eachk∈X^∗. This clearly impliesV(Q)ξ(H0)∈Cforξ∈Cc(R).

The set ofψ∈C0(X^∗) such thatV(Q)ψ(P)∈Cis norm closed and contains all the functions of the form ψ(P) = ξ(h(P +k)) with ξ ∈ Cc(R) and k ∈ X^∗. The family consisting of such functions is a∗-subalgebra ofC0(X^∗) which separates the points ofX^∗ (because|h(p+k)| → ∞ifk→ ∞). By the Stone- Weierstrass theorem, we see that this family is dense in C0(X^∗). So we have V(Q)ψ(P) ∈C ∀ψ ∈C0(X^∗). Here V is an arbitrary function in A^∞. Since A^∞ is dense inA, we finally obtainϕ(Q)ψ(P)∈C for allϕ∈ A, ψ ∈C0(X^∗).

♦

Corollary 2.4 Let h : X^∗ → R be an elliptic polynomial of order m. Then the C^∗-algebra of operators on H(X) generated by the self-adjoint operators H = h(P) +W, where W is a symmetric differential operator of order < m with coefficients inA^∞, is equal toAoX.

Proof. IfV ∈ A^∞, and if we identify X^∗=Rⁿ, we have h(P+k) +V(Q) =h(P) + X

|α|≥1

k^α

α!h^(α)(P) +V(Q)≡h(P) +W so we may use the preceding theorem.

3 Graded C

-Algebras

In this paper we are interested inC^∗-algebras which are graded by a semilattice L. The case when L is finite is presented in Section 8.4 from [1]. Below we extend the formalism to the case of infiniteL. Note that in the present context it is convenient to interchange the rˆoles of the lower and upper bounds in the definition of the grading, which explains some differences in notations with respect to [1].

LetLbe an arbitrarysemilattice, i.e. Lis a partially ordered set such that the lower bounda∧bexists for alla, b∈ L. IfLhas a least or a greatest element, we denote it minLor maxLrespectively. We denote byF(L) the family of finite subsetsF ⊂ Lsuch thata∧b∈ F ifa, b∈ F (the empty set belongs to F(L)).

So eachF ∈F(L) is a finite semilattice for the order relation induced by L, in particular it has a least element minF. We equipF(L) with the order relation given by inclusion. SinceF1∩F2∈F(L) ifF1,F2∈F(L), the setF(L) becomes a semilattice. Note that for each finite part F ⊂ L the set of elements of the form a1∧ · · · ∧an witha1,· · ·, a_n ∈F belongs toF(L) and containsF.

AL-gradedC^∗-algebrais aC^∗-algebraC equipped with a linearly indepen- dent family {C(a)}_a∈L ofC^∗-subalgebras such that:

(i)C(a)·C(b)⊂C(a∧b) for alla, b∈ L (ii) ifF ∈F(L) thenC(F) :=P

a∈FC(a) is a closed subspace ofC (iii)S

F∈F(L)C(F)≡P

a∈LC(a) is dense inC.

It is clear that for each F ∈ F(L) the space C(F) is a C^∗-subalgebra of C and is equipped with a canonical structure of F-gradedC^∗-algebra. The set of C^∗-subalgebras of C (ordered by inclusion) is a semilattice and F 7−→ C(F) is an injective morphism of semilattices (i.e. C(F1∩ F2) = C(F1)∩C(F2)).

In particular, we have C(F₁) ⊂ C(F₂) if F₁ ⊂ F₂ and C coincides with the inductive limit of{C(F) :F ∈F(L)}, a directed system ofC^∗-algebras.

We use the notationC(F) =P

a∈FC(a) for an arbitrary subsetFofL. IfF is∧-stable thenC(F) is a∗-algebra, but is not complete in general. In particular C is just the closure of the∗-algebraC(L). Fora ∈ L set La ={b | a≤ b}, L⁰_a={b|a6≤b}and letCa,Jabe the closure ofC(La) andC(L⁰_a) respectively.

Note thatC(La) is a ∗-subalgebra ofC andC(L⁰_a) is a self-adjoint ideal inC. Hence Ca, Ja are C^∗-subalgebras of C andJa is also a closed self-adjoint ideal in C.

Theorem 3.1 For all a∈ L one has C =Ca+Ja and Ca∩Ja ={0}. The projectionPa:C→Ca determined by this linear direct sum decomposition is a morphism, in particular kPak= 1.

Proof. We clearly haveC(L) =C(La) +C(L⁰_a) as a linear direct sum. LetP_a^◦ be the projection ofC(L) ontoC(La) determined by this decomposition. Thus, ifT ∈C(L) is given byT =P

b∈LT(b), whereT(b)6= 0 only for a finite number ofb, we haveP_a^◦[T] =P

a≤bT(b). ThenP_a^◦[T]^∗=P_a^◦[T^∗] and ifS=P

b∈LS(b) withS(b)6= 0 only for a finite number ofb, then

ST =X

b,c

S(b)T(c) =X

d∈L

X

b∧c=d

S(b)T(c). Thus

P_a^◦[ST] = X

a≤d

X

b∧c=d

S(b)T(c) = X

a≤b∧c

S(b)T(c)

= X

a≤b, a≤c

S(b)T(c) =P_a^◦[S]P_a^◦[T].

HenceP_a^◦ is a morphism of the∗-algebraC(L) onto its ∗-subalgebraC(La).

Let F ∈F(L) such that a∈ F and T(b) = 0 if b /∈ F (we saw before that such a F exists). Then P_a^◦|C(F) is a morphism of the C^∗-algebra C(F) onto theC^∗-algebraC(Fa), with Fa =F ∩ La. Such a morphism always has norm

≤1. Hence we havekP_a^◦[T]k ≤ kTk. Since this is valid for eachT ∈C(L) and C(L) is dense in C, we see that P_a^◦ extends to a morphismPa :C→Ca with Pa[T] =T ifT ∈Ca. In particular,Pa is also a linear projection ofC ontoCa

withkPak ≤1. SinceCa6={0}(becauseC(a)6={0}) we have in factkPak= 1.

We have C(L⁰_a) = kerP_a^◦ ⊂ kerPa. Since C(L) is dense in C, C(L⁰_a) = (1− Pa)C(L), and 1− Pa is a continuous surjective map ofC onto kerPa, we get thatC(L⁰_a) is dense in kerPa. So kerPa=Ja. ♦

One can reformulate the preceding theorem in the following terms: the map P_a^◦:C→Cdefined byP_a^◦[P

bT(b)] =P

a≤bT(b)extends to a norm 1 projection Pa of C onto Ca which is also a morphism. Not also that the family of C^∗- subalgebras{Ca}a∈L is decreasing: ifa≤bthenCb⊂Ca and

PaPb=PbPa=Pb (3.1)

IfLhas a least element minLthenC(minL) is a closed self-adjoint ideal in C, hence one may construct the quotientC^∗-algebra

Cb:=C/C(minL). (3.2)

We shall give a more explicit description of this object when L is an atomic semilattice. We recall that an atom of L is an element a 6= minL such that b ≤a⇒b = minL or b =a. We denote by Mthe set of atoms of L and we say thatLis atomicif eachb6= minLis minorated by an atom. Then we can associate toC a secondC^∗-algebra, namely

Ce:= M

a∈M

Ca (3.3)

where the direct sum is in theC^∗-algebra sense. Observe that there is a natural morphismP :C→Ce, namely

P[T] = (Pa[T])_a∈M.

Theorem 3.2 Assume that the semilatticeLhas a least element and is atomic.

Then the kernel of the morphismP is equal toC(minL).

Proof. First we note that the result is known (and easy to prove) ifLis finite, see Theorem 8.4.1 in [1]; this particular case will be needed below. Clearly Pa[T] = 0 ifT ∈C(minL) anda6= minL, soC(minL)⊂KerP. Reciprocally, let T ∈ C such that Pa[T] = 0 for all a ∈ M. Then for each > 0 there

is F ∈ F(L) and there is S ∈ C(F) such that kT −Sk ≤ . We assume, without loss of generality, that minL ∈ F, hence F is a finite semilattice with minF= minL. Ifb6= minL is an element ofF then there is a∈ Msuch that a≤b, hencePb[T] =PbPa[T] = 0. We get

kP_b[S]k=kP_b[T−S]k ≤ kT−Sk ≤.

Here Pb is the projection associated to the algebraC. However, it is clear (see the remark after the proof of Theorem 3.1) that the restriction of Pb toC(F) coincides with the canonical projection of theF-graded algebraC(F) onto its subalgebraC(Fb),Fb={c∈ F :b≤c}.

LetN be the set of atoms ofF. Then according to Theorem 8.4.1 from [1], the map U 7→ (Pb[U])_b∈N, sending C(F) into L

b∈NC(Fb), has C(minF) = C(minL) as its kernel. The mapC(F)/C(minL) → L

b∈NC(Fb) will be an isometry and since kPb[S]k ≤ for eachb∈ N, the image ofS in the quotient spaceC(F)/C(minL) has norm≤. From the definition of the quotient norm it follows that there isS0∈C(minL) such thatkS−S0k ≤2(in fact≤).

Thus, we see that for each >0 there isS0∈C(minL) such that kT−S0k=kT−S+S−S0k ≤3.

SinceC(minL) is closed we getT ∈C(minL). ♦

The preceding theorem gives us a canonical embeddingCb ⊂Ce, more pre- cisely

C/C(minL),→ M

a∈M

Ca. (3.4)

Although easy to prove, this result is important: it allows one to compute the essential spectrum and to prove the Mourre estimate under very general assumptions. The range of the map (3.4) can be explicitly described, but this is irrelevant for our purposes.

4 C

-Algebras Associated to Subspaces

4.1

Let X be a finite-dimensional real vector space andY a linear subspace. We denote πY =π^X_Y the canonical surjection of X onto the quotient vector space X/Y and Y^⊥ the set of x^∗ ∈ X^∗ such that hy, x^∗i = 0 ∀y ∈ Y. We have canonical identifications (X/Y)^∗=Y^⊥ andX^∗/Y^⊥=Y^∗.

We shall embedC0(X/Y)⊂Cb(X) with the help of the mapϕ7−→ϕ◦πY. SinceCb(X)⊂B(X), we shall have

C0(X/Y)⊂Cbu(X/Y)⊂Cbu(X)⊂B(X). (4.1) Forϕ∈Cb(X/Y) we shall denote ϕ(QY) = (ϕ◦πY)(Q) the operator inB(X) associated to it. Sometimes it is important to specify in the notations the space X; then we setϕ(QY) =ϕ(Q^X_Y).

The relationY^∗=X^∗/Y^⊥ implies

Cb(Y^∗) =Cb(X^∗/Y^⊥)⊂Cb(X^∗)⊂B(X). (4.2) For ψ∈ Cb(Y^∗) we denote ψ(PY) or ψ(P_Y^X) the operator inB(X) associated to it; we have ψ(P_Y^X) = Φ_X∗[ψ(Q^X_Y⊥^∗]. Observe, in particular, that the group C^∗-algebraC0(Y^∗) of the additive groupY is embedded inB(X).

LetG(X) be the Grassmannian ofX, i.e. the lattice of all vector subspaces of X with inclusion as order relation. Note that for Y, Z ∈ G(X) one has Y∧Z =Y∩ZandY∨Z=Y+Z. For eachY ∈G(X) we have aC^∗-subalgebra C0(X/Y) ofCb(X) as explained above. In particular C0(X/O) =C0(X) and C0(X/X) =C. Note that eachC0(X/Y) is translation invariant, i.e. it is stable under all the automorphismsτx,x∈X.

IfF ⊂G(X) is a family of vector subspaces ofX then we set C₀^X(F) = X

Y∈F

C0(X/Y). (4.3)

This is the linear subspace ofCbu(X) generated byS

Y∈FC0(X/Y). Note that C₀^X(∅) ={0}and C₀^X(Y)≡C₀^X({Y}) =C0(X/Y).

Lemma 4.1 (a) The family {C0(X/Y) : Y ∈ G(X)} of C^∗-subalgebras of Cb(X)is linearly independent.

(b)If F ⊂G(X)is finite thenC₀^X(F)is a closed subspace of Cbu(X).

(c) For each Y, Z ∈G(X)the set C0(X/Y)·C0(X/Z)is a dense subalgebra of C0(X/(Y ∩Z)).

Proof. We give a detailed proof of this simple lemma because the same ar- gument will be used later on in order to prove Theorem 4.5. Let F ⊂ G(X) be finite and for eachY ∈ F letϕY ∈C0(X/Y). Denoteϕ^Y :X/Y →C the function such that ϕY =ϕ^Y ◦πY. Then for eachω ∈X one has (τωϕY)(x) = ϕ^Y(πY(x) +πY(ω)) for allx∈X. Hence, if we setFω={Y ∈ F:ω∈Y}, then

λ→∞lim τλω[X

Y∈F

ϕY] = X

Y∈Fω

ϕY (4.4)

pointwise onX. In particular k X

Y∈Fω

ϕYk ≤ kX

Y∈F

ϕYk (4.5)

wherek · kis the sup norm.

Let us prove that there is a numberC such that for allZ∈ F and all{ϕY} as above

kϕZk ≤Ck X

Y∈F

ϕYk. (4.6)

This clearly implies (a) and (b). If the set Z0=Z\ [

Y6=Z

Y = \

Y6=Z

[Z\(Y \ Z)]

is not empty then (4.5) with a choice ω ∈ Z0 gives (4.6) (withC = 1). Since Y ∩Z are linear subspaces ofZ one hasZ0 =∅ if and only if there isY ∈ F such that Z⊂Y strictly. This cannot happen ifZ is a maximal element inF, hence (4.6) holds for such elements. LetF₁ be the set ofY ∈ F which are not maximal elements in F. Then we clearly get kP

Y∈F1ϕYk ≤ C1kP

Y∈FϕYk for some constantC1. By what we already proved we see then that (4.6) holds for the maximal elementsZ ofF₁, etc.

We now prove (c). Let E = (X/Y)×(X/Z) equipped with the direct sum vector space structure. If ϕ ∈ C0(X/Y) and ψ ∈ C0(X/Z) then ϕ⊗ψ denotes the function (s, t)7−→ϕ(s)ψ(t), which belongs toC0(E). The subspace generated by the functions of the form ϕ⊗ψ is dense inC0(E) by the Stone- Weierstrass theorem. Let F be a linear subspace of E. Since each function in C0(F) extends to a function in C0(E) we see that the restrictions (ϕ⊗ψ)|F generate a dense linear subspace ofC0(F).

Let us denote byπthe mapx7−→(πY(x), πZ(x)), soπis a linear map from X to E with kernel V = Y ∩Z. Let F be the range of π. Then there is a linear bijective map ˜π : X/V → F such that π = ˜π◦πV. So θ 7−→ θ◦˜π is an isometric isomorphism of C0(F) onto C0(X/V). Hence for ϕ ∈ C0(X/Y) andψ∈C0(X/Z) the functionθ= (ϕ⊗ψ)◦˜πbelongs toC0(X/V), it has the propertyθ◦πV =ϕ◦πY·ψ◦πZ, and the functions of this form generate a dense

linear subspace ofC0(X/V). ♦

We say that F ⊂ G(X) is ∩-stable ifY, Z ∈ F ⇒ Y ∩Z ∈ F (so F is a generalized flag of subspaces of X). Such aF is a semilattice when equipped with the order relation given by inclusion. We denote by F(X) =F(G(X)) the set of finite ∩-stable subsets ofG(X).

Corollary 4.2 IfF ∈F(X)thenC₀^X(F)is aC^∗-subalgebra ofCbu(X)equipped with a natural structure of F-graded C^∗-algebra. This algebra is unital if and only if X ∈ F. For F1,F2∈F(X)one has

C₀^X(F1)\

C₀^X(F2) =C₀^X(F1

\F2). (4.7)

In particular, one has C₀^X(F1)⊂C₀^X(F2)if and only if F1⊂ F2.

4.2

We are ready to define the noncommutative versions of the algebrasC0(X/Y):

they are cross products of algebras of the formC0(X/Y) by the natural actionτ of the additive groupX. These algebras have been first introduced, in a rather different form, by Perry, Sigal and Simon in [PSS]. The connection between our formulation and theirs is clarified by Proposition 4.7 below. See also the introductions of chapters 8 and 9 in [1].

Definition 4.3 If Y is a subspace of X then C₀^X(Y) = C0(X/Y)oX is the C^∗-subalgebra of B(X) obtained as norm closure of C0(X/Y)·C0(X^∗). For each subsetF ⊂ G(X)letC₀^X(F) be the linear subspace ofB(X)generated by the algebras C₀^X(Y) withY ∈ F, so

C₀^X(F) = X

Y∈F

C₀^X(Y). (4.8)

Observe that C₀^X(F) = [[C₀^X(F)·C0(X^∗)]] = [[C0(X^∗)·C₀^X(F)]]. So for

∩-stableF we haveC₀^X(F) =C₀^X(F)oX.

To each vector subspace Y of X we have thus associated a C^∗-subalgebra C₀^X(Y) ofB(X). The only one which is abelian is

C₀^X(X) =C0(X^∗) ={ϕ(P)|ϕ∈C0(X^∗)}. (4.9) The algebraC₀^X(O) is generated byC0(X)·C0(X^∗) and, since the operators of the formϕ(Q)ψ(P)) withϕ∈C0(X), ψ∈C0(X^∗) are compact, we have

C₀^X(O) =K(X). (4.10)

The algebras which play the main rˆole in theN-body problem (as presented in ch. 9 of [1]) are of the formC₀^X(F) with finite F and will be studied in this section. The next one is devoted to the caseF=G(X).

We shall need an extension of the automorphismτxofCb(X) to an automor- phism ofB(X): we setτx[S] = e^ihx,PiSe^−ihx,Pi for eachx∈X and S∈B(X).

Observe that forϕ∈C0(X/Y) andψ∈C0(X^∗) one has τx[ϕ(QY)ψ(P)] =ϕ(QY +πY(x))ψ(P). This immediately gives the next lemma.

Lemma 4.4 (i)If y∈Y andS∈C₀^X(Y)thenτy[S] =S;

(ii) if S ∈ C₀^X(Y) and πY(x) → ∞ then τx[S] → 0 in the strong operator topology.

Theorem 4.5 (a)The family{C₀^X(Y) :Y ∈G(X)}ofC^∗-subalgebras ofB(X) is linearly independent.

(b)If F ⊂G(X)is finite thenC₀^X(F)is(norm)closed in B(X).

(c) If Y, Z ∈G(X)then C₀^X(Y)·C₀^X(Z) is a dense linear subspace of the C^∗- algebra C₀^X(Y ∩Z).

Proof. LetF ⊂G(X) be finite and for eachY ∈ F letT(Y)∈C₀^X(Y). Then s− lim

λ→∞τλω[X

Y∈F

T(Y)] = X

Y∈Fω

T(Y) (4.11)

where the notations are as in the proof of Lemma 4.1. Indeed, this is an im- mediate consequence of (i) and (ii) above. Now (a) and (b) follow by the same argument as in Lemma 4.1.

We shall deduce (c) from the corresponding assertion of Lemma 4.1. By Corollary 2.2C₀^X(Y) is the norm closed linear space generated by the operators of the form ψY(P)ϕY(QY), with ψY ∈ C0(X^∗) and ϕY ∈ C0(X/Y). On the other hand C₀^X(Z) is, by definition, the norm closure of the linear space gen- erated by the operators of the form ϕZ(QZ)ψZ(P), with ϕZ ∈C0(X/Z) and ψZ ∈ C0(X^∗). By Lemma 4.1 one has ϕY(QY)ϕZ(QZ) = ϕV(QV) for some ϕV ∈C0(X/V), whereV =Y ∩Z. So

ψY(P)ϕY(QY)·ϕZ(QZ)ψZ(P) =ψY(P)·ϕV(QV)ψZ(P)

which clearly belongs to C₀^X(Y ∩Z). This proves that C₀^X(Y)·C₀^X(Z) ⊂ C₀^X(Y ∩Z).

Elements of the formψ1(P)ϕV(QV)ψ2(P), withψ1,ψ2∈C0(X^∗) andϕV ∈ C0(X/V), clearly generate C₀^X(Y ∩Z) (because those elements of the form ϕV(QV)ψ1(P)ψ2(P) do and we may use Lemma 2.1). Hence the density of C₀^X(Y)·C₀^X(Z) inC₀^X(Y ∩Z) follows immediately from Lemma 4.1. ♦The following result is an immediate consequence of Theorem 4.5

Theorem 4.6 If F ∈ F(X) then C₀^X(F) is a C^∗-subalgebra of B(X). If we equipF with the order relation given by inclusion then the family{C₀^X(Y)}Y∈F

of C^∗-subalgebras of C₀^X(F) provides C₀^X(F)with a structure of F-gradedC^∗- algebra.

4.3

The choice of a supplementary subspaceZofY inX will give us a canonical iso- morphism between C(X/Y) and theC^∗-tensor product of the algebrasC0(Y^∗) andK(Z):

C(X/Y)∼=C0(Y^∗)⊗K(Z). (4.12) In order to define in a precise way this isomorphism let us introduce some notations. SinceX =Y +Z (direct sum) we have a canonical identification of X^∗withY^∗⊕Z^∗. LetiY, iZ be the inclusion maps ofY, Z intoX respectively.

By taking adjoints we get an isomorphism (i^∗_Y, i^∗_Z) :X^∗→Y^∗⊕Z^∗and we may define for any functionsu:Y^∗→Candv:Z^∗→Cthe functionu⊗v:X^∗→C by (u⊗v)(x^∗) = u(i^∗_Yx^∗)v(i^∗_Zx^∗). If u ∈ C0(Y^∗), v ∈ C0(Z^∗) then clearly u⊗v∈C0(X^∗). Moreover, the projectionpZ :X →Zdetermined by the direct sum decompositionX =Y+Zfactorizes to an isomorphismp^[_Z:X/Y →Zand if forw:Z →Cwe definew^[:X/Y →Cbyw^[=w◦p^[_Z, thenw^[∈C0(X/Y) ifw∈C0(Z).

Proposition 4.7 There is a linear continuous mapC0(Y^∗)⊗K(Z)→C₀^X(Y) such that for each u∈C0(Y^∗), v∈C0(Z^∗)andw∈C0(Z)the elementu(P^Y)⊗

w(Q^Z)v(P^Z)

is sent into w^[(QY)(u⊗v)(P^X). This map is uniquely defined and is an isomorphism.

Proof. The uniqueness of the map and its surjectivity follow immediately from the fact that the elements of the formu(P^Y)⊗

w(Q^Z)v(P^Z)

andw^[(QY)(u⊗ v)(P^X) span dense linear subspaces inC0(Y^∗)⊗K(Z) andC₀^X(Y) respectively.

To prove the existence and the isomorphism properties observe that we get an isomorphismJ :H(X) → H(Y)⊗ H(Z) by setting (Jf)(y, z) =f(y+z). It remains to check that

u(P^Y)⊗

w(Q^Z)v(P^Z)

· J =J ·w^[(QY)(u⊗v)(P^X)

which is a straightforward consequence of the definitions. ♦ The preceding tensor product decomposition of H(X) also gives canoni- cal isomorphisms C0(X/Y) ∼= 1⊗C0(Z) and C0(X^∗) ∼= C0(Y^∗)⊗C0(Z^∗).

This induces a linear isomorphism of the vector spacesC0(X/Y)·C0(X^∗) and C0(Y^∗)⊗[C0(Z)·C0(Z^∗)] which extends to the isomorphism between C₀^X(Y) andC0(Y^∗)⊗K(Z) indicated above.

If X is equipped with a scalar product α and if H(X) is identified with H(Y)⊗ H(Y_α^⊥), where Y_α^⊥ is the orthogonal space of Y in X, then C₀^X(Y) will be equal toC0(Y^∗)⊗K(Y_α^⊥). The algebrasC0(Y^∗) andC0(Y^∗)⊗K(Y_α^⊥) are denoted T(Y) andT(Y_α^⊥) in [1]. If αis replaced by a new scalar product β, so that Y has a different orthogonal subspace Y_β^⊥, then C0(Y^∗)⊗K(Y_β^⊥) gives (after the identification H(X) = H(Y)⊗ H(Y_β^⊥)) the same algebra as C0(Y^∗)⊗K(Y_α^⊥) (cf. Proposition 4.7). So the algebra T(Y_α^⊥) is determined by Y, independently of any Euclidean structure on X. Our present notation C₀^X(Y) stresses this fact.

5 The Algebra

5.1

In this section we shall study the C^∗-algebra

C₀^X := norm closure inB(X) of C₀^X(G(X)). (5.1) For this we apply in the present context the general theory of Section 3: we take L = G(X) and C(Y) = C₀^X(Y) for Y ∈ G(X). So the algebra C₀^X is G(X)-graded and can be identified with the inductive limit of the family of C^∗-algebras{C₀^X(F)| F ∈ F(X)}. Indeed, if we orderF(X) by the inclusion relation thenC₀^X(F1)⊂C₀^X(F2) if and only ifF1⊂ F2and

[

F∈F(X)

C₀^X(F) =C₀^X(G(X)) (5.2) is a dense∗-subalgebra ofC₀^X. Moreover, for allF1,F2 inF(X) we have

C₀^X(F₁)∩C₀^X(F₂) =C₀^X(F₁∩ F₂). (5.3) If Y ∈ G(X) then we denote by C_Y^X and J_Y^X the norm closures in B(X) of the spacesP

Y⊂ZC₀^X(Z) andP

Y6⊂ZC₀^X(Z) respectively. Observe that the

notations are consistent: ifY =O={0}thenC_O^X=C₀^X. In the next theorem, which is an immediate consequence of Theorem 3.1, we point out a canonical projection (in the sense of linear spaces) ofC₀^X onto its subspaceC_Y^X.

Theorem 5.1 C_Y^X is aC^∗-subalgebra ofC₀^X, J_Y^X is a closed self-adjoint ideal in C₀^X, andC₀^X is equal to their direct sum: C₀^X =C_Y^X+J_Y^X andC_Y^XT

J_Y^X= {0}. The linear projection PY of the linear space C₀^X onto its linear subspace C_Y^X determined by the preceding direct sum decomposition is a morphism (in particular it is an operator of norm 1).

The family{C_Y^X |Y ∈G(X)} ofC^∗-subalgebras ofC₀^X is decreasing

Y ⊂Z⇒CZ^X ⊂CY^X (5.4)

and has a least element C_X^X =C0(X^∗). Clearly (5.4) implies

PZPY =PYPZ =PZ. (5.5)

Our purpose now is to describe the quotient of the algebraC₀^X with respect to the ideal C₀^X(O) = K(X) of compact operators. The next result is an immediate consequence of Theorem 3.2.

Theorem 5.2 Let P(X)be the projective space associated to X, i.e. the set of all one dimensional subspaces of X. Denote by Ce₀^X the C^∗-direct sum of the algebras C_Y^X withY ∈P(X):

Ce₀^X = M

Y∈P(X)

C_Y^X. (5.6)

Let P :C₀^X→Ce₀^X be defined by

P[T] = M

Y∈P(X)

PY[T]. (5.7)

Then P is a morphism and its kernel is equal toK(X).

SoP induces an embedding of theC^∗-algebraC₀^X/K(X) intoCe₀^X. We shall identify C₀^X/K(X) with a subalgebra ofCe₀^X:

C₀^X/K(X)⊂Ce₀^X. (5.8)

5.2

We shall make here some final remarks concerning the algebra C₀^X. First we give another description of the mapsPY. Observe that by Theorem 4.5(a) each T ∈C₀^X(G(X)) can be written in a unique way as a sumT =P

Z∈FT(Z) with F ⊂G(X) finite andT(Z)∈C₀^X(Z),T(Z)6= 0. For such a T we have

PY[T] = X

Z∈F,Z⊃Y

T(Z) (5.9)

and this property uniquely characterizes PY. If ω ∈ Y is such that ω /∈ Z if Z∈ F andY 6⊂Z (such a choice is possible becauseF is finite), then

s− lim

λ→∞τλω[T] = X

Z⊃Y

T(Z), by Lemma 4.4. In other terms, forT as above we have

PY[T] =s− lim

λ→∞τλω[T]. (5.10)

In particular we get:

Lemma 5.3 If Y is a one dimensional subspace of X and ω ∈ Y \ {0} then one has for allT ∈C₀^X:

PY[T] =s− lim

λ→∞τλω[T]. (5.11)

In particular, we see that the main assertion of Theorem 5.2, namely the relation KerP =K(X), is equivalent to the following one: for T ∈C₀^X one has T ∈K(X)if and only if w−lim_λ→∞τλω[T] = 0for each ω∈X\ {0}.

Notice that there is an abelian version of the algebraC₀^X, namely the closure C₀^X in Cb(X) ofP

Y⊂XC0(X/Y), and everything we have done applies to C₀^X too. In particular, forf ∈C₀^X we have: f ∈C0(X) if and only if lim_λ→∞f(x+ λω) = 0 for eachω ∈ X\ {0}. A geometric proof of this not obvious fact (if dimX > 2) has been shown to us by Radu-Alexandru Todor. We thank him for that.

Certain partitions of unity introduced by Froese and Herbst in [3] have proved to be very useful in the usual treatment ofN-body Hamiltonians. We shall briefly present them and their relation with the algebrasC₀^X(F). Below we assume that a Euclidean norm is given on X.

Let χ : X → R be a C^∞ function, homogeneous of degree zero outside the unit sphere. Since the algebra C₀^X is generated by functions of the form ϕ(Q)ψ(P) (orψ(P)ϕ(Q)) withψ∈ S(X^∗), it is easy to prove that [χ(Q), T]∈ K(X) for allT ∈C₀^X.

Now let Z ⊂X be a subspace and assume thatχ(z) = 0 ifz ∈Z,|z| ≥1.

Then for eachϕ∈C0(X/Z) the functionχ·ϕ◦πZ belongs toC0(X) (indeed, if x→ ∞andπZ(x) is bounded, thenx/|x|approachesZ), henceχ(Q)ϕ(QZ)ψ(P) is a compact operator. It follows thatχ(Q)T andT χ(Q) are compact operators ifT ∈C₀^X(Z).

Let us fixF ∈F(X) withO, X∈ F and letY ∈ F, Y 6=X. AC^∞function χ_Y :X →Rwhich is homogeneous of degree zero on|x| ≥1 is called (according to Froese-Herbst)Y-reducing if: for eachZ ∈ F with Y 6⊂Z and each z∈Z,

|z| ≥1, one hasχ_Y(z) = 1. By what we said above, we see that (i) [χ_Y(Q), T]∈K(X)∀T ∈C(F)

(ii)χ_Y(Q)TandT χY(Q) belong toK(X) ifT ∈C(F_Y⁰ ) =P

Z∈F,Y6⊂ZC₀^X(Z).

LetN be the set of atoms ofF. An F-reducing partition of unity on X is a family{χ_Y}Y∈N such thatχ_Y isY-reducing andP

Y∈Nχ²_Y = 1 onX. In [3]

such families are constructed. From (i) and (ii) above we then get: ifS∈C(F) and if we denote SY =PY[S] its canonical projection onto C(FY), then there isK∈K(X) such that

S=K+ X

Y∈N

χ_Y(Q)SY χ_Y(Q). (5.12) It is clear thatF cannot be replaced by an infinite semilattice in the preced- ing construction. However, these partitions can be used to give an alternate and more elementary proof of the main assertion of Theorem 5.2, namely thatT is compact ifT ∈C₀^X and w−lim_λ→∞τλω[T] = 0 for eachω ∈X\ {0}. Indeed, for each >0 we can findF as above and S ∈C(F) such that kT −Sk ≤ . Note that we can assume T andS self-adjoint. Write S =P

{S(Z) :Z ∈ F}

with S(Z) ∈ C₀^X(Z), hence SY = P

{S(Z) : Z ∈ F, Z ⊃ Y}, and let Fω = {Z ∈ F : ω ∈ Z} for ω ∈ X\ {0}. Then s−lim_λ→∞τλω[S] = P

Z∈FωS(Z) and from the Fatou lemma we getkP

Z∈FωS(Z)k ≤.For eachY ∈ N we can find ω ∈Y \ {0} such that ω /∈Y⁰ ifY⁰ ∈ N \ {Y}. Hence we getkSYk ≤ , or − ≤ SY ≤ , for each Y ∈ N. Then − ≤ P

χ_Y(Q)SYχ_Y(Q) ≤ be- causeP

χ_Y(Q)²= 1. So from (5.12) we see that there isK∈K(X) such that kS−Kk< . This implies kT−Kk ≤2, which proves the assertion.

References

[1] W. Amrein, A. Boutet de Monvel, and V. Georgescu,C₀-Groups, Commu- tator Methods and Spectral Theory of N-Body Hamiltonians, Birkh¨auser, Progress in Math. Ser.,135, 1996.

[2] M. Damak and V. Georgescu,C^∗-Algebras Related to theN-Body Problem and Self-Adjoint Operators Affiliated to Them, to be submitted.

[3] R. G. Froese and I. Herbst, A New Proof of the Mourre Estimate, Duke Math. Journal 49, 1075-1085 (1982).

[4] V. Georgescu and A. Iftimovici, C^∗-Algebras of Energy Observables, in preparation.

[5] P. Perry, I.M. Sigal and B. Simon,Spectral Analysis ofN-Body Schr¨odinger Operators, Ann. Math. 114 (1981), 519-567.

Mondher Damak & Vladimir Georgescu CNRS (ESA 8080) and Department of Mathematics University of Cergy-Pontoise

2, avenue Adolphe Chauvin

95302 Cergy-Pontoise Cedex, France

e-mail: [email protected]