Mathematical Physics and Quantum Field Theory, Electronic Journal of Differential Equations, Conf. 04, 2000, pp. 51–69 http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

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^{∗}- algebra C_{0}^{X} graded by the lattice of all subspaces of X. 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 with C_{0}^{X}.

1 Introduction

Let C be a C^{∗}-algebra of bounded operators on a Hilbert space H and let H be a self-adjoint operator on H. One says that H is affiliated to C if (H −z)^{−1} ∈ C for some z ∈ ℂ \ σ(H). In this case φ(H) ∈ C for all φ ∈ C_{0}(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 quotient C^{∗}-algebra Ĉ = C/K(H), and one can consider the map Ĥ : C_{0}(ℝ) → Ĉ defined by Ĥ(φ) = { \atop φ(H)} = [class of φ(H) in the quotient]. This is a morphism, and Ĥ should be thought as an “abstract” self-adjoint operator affiliated to the “abstract” C^{∗}-algebra Ĉ. The notion of spectrum of Ĥ has an obvious meaning, and it is easy to show that the essential spectrum of H is given by σ_{ess}(H) = σ(Ĥ).

This point of view is not practically useful unless the algebra Ĉ has a special structure. In any case Ĉ should be, in some sense, simpler than C. A systematic treatment of a rather large class of examples in which Ĉ and Ĥ 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.

52 C^{∗}-cross products

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 the N-body problem it is natural (at least, a posteriori) to consider in this case the algebra C = C_{0}^{X} constructed as follows. Let X be a real finitedimensional vector space (the configuration space of the system of N particles) and A the C^{∗}-algebra generated by functions on X of the form φ ∘ π_{Y}, where Y is a vector subspace of X, φ ∈ C_{0}(X/Y) and π_{Y} : X → X/Y is the natural map. The additive group X acts continuously by translations on A. Then we take C equal to the cross product of A by the action of X.

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

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

\begin{matrix}c \\ ◯ \end{matrix}2000 Southwest Texas State University and University of North Texas.

Published July 12, 2000.

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 usefulness in more complicated situations is, however, shown in [4]). Let H(X) be the space of square integrable functions on X and denote by Q and P the position and momentum observables. If A is a ∗-algebra of bounded uniformly continuous functions on X and if A is stable under translations, then the (norm) closed linear subspace of B(H(X)) generated by operators of the form φ(Q)ψ(P), with φ ∈ A, ψ ∈ C_{0}(X^{∗}), is a C^{∗}-algebra isomorphic to the “abstract” cross product A ⋊ X of A by the action of the additive group X. Here we simply take this as the definition of A ⋊ X. Thus the algebra C_{0}^{X} is obtained by choosing A equal to the closed linear space generated by functions of the form φ ∘ π_{Y} with φ ∈ C_{0}(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 = H_{0} + V, where H_{0}, the kinetic energy, is an elliptic differential operator of order m with constant coefficients and V is a symmetric differential operator of order < m with coefficients in A. Assume also that A contains the constant functions. Then the smallest C^{∗}-algebra of operators on H(X) to which all these operators are affiliated (H_{0} being fixed) is equal to C = A ⋊ X. The main point of our approach is that this algebra often has a remarkable structure (determined by certain properties of A), and this fact alone gives important information on the spectral properties of H. Note that we construct C starting with a rather restricted class of perturbations V. However, the class of operators H affiliated to C 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 that X = X^{∗} = ℝ^{n}. 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 of N-body type. Indeed, one can replace X 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 Let H be a self-adjoint operator on H(X) affiliated to C_{0}^{X}. Then for each ω ∈ X, ω ≠ 0, the limit lim_{|λ|→∞} τ_{λω}[H] := H_{ω} exists in strongresolvent sense and

σ_{ess}(H) = \overline{⋃_{ω∈X\\{0\}} σ(H_{ω})}.  (1.1)

Here τ_{x}[H] = e^{i⟨x,P ⟩}He^{−i⟨x,P ⟩} and (e^{i⟨x,P ⟩}f)(y) = f(x + y). Note that H_{ω} 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^{∗} = ℝ^{n} and let h : ℝ^{n} → ℝ be a continuous function such that c^{−1}⟨x⟩^{2s} ≤ h(x) ≤ c⟨x⟩^{2s} if |x| > r, where s, r, c are strictly positive constants and ⟨x⟩ = (1+|x|^{2})^{1/2}. Then H_{0} = h(P) is a self-adjoint operator whose form domain is the Sobolev space H^{s}(X). For each linear subspace Y ⊂ X let V_{Y} be a continuous symmetric sesquilinear form on H^{s}(X), identified with an operator H^{s}(X) → H^{−s}(X), such that:

(i) [e^{i⟨y,P ⟩}, V_{Y}] = 0 for all y ∈ Y;

(ii) ∥[e^{i⟨Q,k⟩}, V_{Y}]∥_{H^{s}→H^{−s}} → 0 if k ∈ Y, k → 0

(iii) ∥[e^{i⟨Q,k⟩} − 1]V_{Y} ∥_{H^{s}→H^{−s}} → 0 if k ∈ Y^{⊥}, k → 0.

We have denoted e^{i⟨Q,k⟩} the operator of multiplication by the function e^{i⟨x,k⟩}. For Y = O = {0} we take V_{Y} = 0. Note that in (ii) (but not in (iii)) the Euclidean structure of X = ℝ^{n} is used. Furthermore, we assume

∑_{Y ⊂X} ∥V_{Y} ∥_{H^{s}→H^{−s}} < ∞,

in particular V_{Y} ≠ 0 only for a countable number of subspaces Y. Finally, we ask that V_{Y} ≥ −μ_{Y} H_{0} − δ_{Y} as forms on H^{s}(X), where μ_{Y} , δ_{Y} ≥ 0 are numbers such that ∑_{Y} μ_{Y} < 1 and ∑_{Y} δ_{Y} < ∞. Then H = H_{0}+∑_{Y} V_{Y} is a self-adjoint, bounded from below operator on H(X), H is affiliated to C_{0}^{X}, and

σ_{ess}(H) = \overline{⋃_{Y ∈M} σ(H_{Y})}.

Here H_{Y} = H_{0} + ∑_{Z⊃Y} V_{Z} and M is the set of minimal elements (for the inclusion relation) of the class of subspaces of the form Y_{1} ∩ ⋯ ∩ Y_{k} ≠ 0 with Y_{i} ∈ {Y : V_{Y} ≠ 0}.

Theorem 5.2 can also be used in order to prove the Mourre estimate for operators affiliated to C_{0}^{X}. We shall present here only the simplest case when H_{0} is the (positive) Laplacian and the V_{Y} do not depend on the projection on Y of the momentum, so V_{Y} = 1 ⊗ V^{Y} if H(X) = H(Y) ⊗ H(Y^{⊥}). More precisely, it suffices to replace (ii) by the stronger condition:

(ii^{∗}) [e^{i⟨Q,k⟩}, V_{Y}] = 0 if k ∈ Y.

Now H looks exactly as in the non-relativistic N-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 of H. Note that in the N-body case the subspaces Y are usually denoted X_{a}; moreover, one must interchange the rôles of Y and Y^{⊥} in order to agree with the conventions from [1].

Theorem 1.2 Let H be of the form described above and let D be the generator of the dilation group in H(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 to C_{0}^{X}. Let L be the family of subspaces of the form Y_{1} ∩ ⋯ ∩ Y_{k} where Y_{i} are subspaces such that V_{Y_{i}} ≠ 0. Define the set of thresholds of H by

τ(H) = \overline{⋃_{Y ∈L\\{O\}} σ_{p}(H^{Y})}.

Then the Mourre estimate for H with respect to D holds outside τ(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 then C_{b}(X) is the C^{∗}-algebra of continuous bounded complex functions on it, C_{0}(X) the C^{∗}-subalgebra of functions convergent to zero at infinity, and C_{c}(X) the subalgebra of functions with compact support. If H is a Hilbert space then B(H) and K(H) are the C^{∗}-algebras of all bounded and compact operators on H respectively. By morphism between two ∗-algebras we mean ∗-morphism. If A, B are subspaces of an algebra C then we denote by A · B the linear subspace of C generated by the elements of the form AB with A ∈ A, B ∈ B; if C is a C^{∗}-algebra then [[A · B]] is the norm closure of A · B in C. A family {C_{i}}_{i∈I} of subalgebras of C is linearly independent if for each family {S_{i}}_{i∈I} such that S_{i} ∈ C_{i} ∀i, S_{i} ≠ 0 for at most a finite number of i and ∑_{i∈I} S_{i} = 0, one has S_{i} = 0 for all i ∈ I.

Let X be a finite-dimensional real vector space. For each x ∈ X we denote by τ_{x} the automorphism of C_{b}(X) associated to the translation by x, so (τ_{x}φ)(y) = φ(y + x). Then the set C_{bu}(X) of functions φ ∈ C_{b}(X) such that x ↦ τ_{x}φ is norm continuous is the C^{∗}-algebra of bounded uniformly continuous functions on X. If X = O = {0} is the zero dimensional vector space then C_{b}(X) = C_{bu}(X) = C_{0}(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 on X). There is no canonical norm on H(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)) and K(H(X)) are independent of the choice of the measure. We set B(X) = B(H(X)) and K(X) = K(H(X)); these are C^{∗}-algebras uniquely determined by X. If X = O = {0} is the zero dimensional vector space we set H(X) = ℂ and B(O) = K(O) = ℂ.

We embed C_{b}(X) in B(X) by identifying a function u ∈ C_{b}(X) with the operator of multiplication by u in H(X). If we denote u(Q), or u(Q^{X}), this operator, then the map u ↦ u(Q) is an isometric morphism of C_{b}(X) into B(X). So we realize C_{b}(X) as a C^{∗}-subalgebra of B(X).

Let X^{∗} be the dual space of X. For x ∈ X, x^{∗} ∈ X^{∗} let ⟨x, x^{∗}⟩ = x^{∗}(x) and identify X^{∗∗} = X by setting ⟨x^{∗}, x⟩ = ⟨x, x^{∗}⟩. We shall embed the algebra C_{b}(X^{∗}) of bounded continuous functions on X^{∗} into B(X) by the following procedure, explicitly independent of a Haar measure on X. It will be convenient here to use the notation e^{i⟨x,P ⟩} for the operator of translation by x ∈ X, more precisely (e^{i⟨x,P ⟩}f)(y) = f(x + y) (this is identical to the action of τ_{x} on functions). For each v ∈ S(X^{∗}) (space of Schwartz test functions on X^{∗}) there is a unique measure v̂ on X such that v(x^{∗}) = ∫_{X} e^{i⟨x,x^{∗}⟩}v̂(dx). We set v(P) = ∫_{X} e^{i⟨x,P ⟩}v̂(dx) (if X has to be specified we set v(P) = v(P^{X})). The map v ↦ v(P) extends to an isometric morphism of C_{0}(X^{∗}) into B(X). This allows us to realize C_{0}(X^{∗}) as a C^{∗}-subalgebra of B(X) and it is easy to check that the action of C_{0}(X^{∗}) on H(X) is nondegenerate. This, in turn, will give us an embedding C_{b}(X^{∗}) ⊂ B(X), uniquely determined by the property u(P)v(P) = (uv)(P) if u ∈ C_{b}, v ∈ C_{0}. Observe that C_{0}(X^{∗}) is just the group C^{∗}-algebra of the additive group X.

One may define the Fourier transformation as a bijective map F : S(X) → S(X^{∗}) such that F exp(i⟨x, P^{X}⟩) = exp(i⟨x, Q^{X^{∗}})⟩F for all x ∈ X. This defines F modulo a complex factor and Φ_{X}[S] = F SF^{−1} gives a canonical isomorphism Φ_{X} : B(X) → B(X^{∗}) such that Φ_{X}[v(P^{X})] = v(Q^{X^{∗}}) for all v ∈ C_{b}(X^{∗}).

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

Lemma 2.1 If u ∈ C_{bu}(X) and v ∈ S(X^{∗}) then for each number ϵ > 0 there are points x_{1}, …, x_{N} ∈ X and functions v_{1}, …, v_{N} ∈ S(X^{∗}) such that

∥v(P)u(Q) − ∑_{k=1}^{N} u(Q + x_{k})v_{k}(P)∥ < ϵ.

Proof. Set S = u(Q) and S(x) = e^{i⟨x,P ⟩}Se^{−i⟨x,P ⟩}. Then S(x) = u(Q + x), S is an element of B(X), and the map x ⟼ S(x) is bounded by ∥S∥ and is norm continuous. Choose a Haar measure dx on X. The measure v̂ is absolutely continuous, so we can identify it with a function in S(X) by v̂(dx) ≡ v̂(x)dx. Then

v(P)S = ∫_{X} e^{i⟨x,P ⟩}Sv̂(x)dx = ∫_{X} S(x)e^{i⟨x,P ⟩}v̂(x)dx.

Let K be a compact subset of X and V a compact neighbourhood of the origin in X. One can find functions J_{0}, J_{1}, …, J_{N} of class C^{∞} on X such that:

(i) 0 ≤ J_{k} ≤ 1 if 0 ≤ k ≤ N and ∑_{k=0}^{N} J_{k}(x) = 1 for all x ∈ X;

(ii) K ∩ supp J_{0} = ∅;

(iii) there are points x_{1}, …, x_{N} ∈ X and such that supp J_{k} ⊂ x_{k} + V if 1 ≤ k ≤ N.

Then we write:

v(P)S = ∑_{k=1}^{N} S(x_{k}) ∫_{X} J_{k}(x)e^{i⟨x,P ⟩}v̂(x)dx

+ ∑_{k=1}^{N} ∫_{X}[S(x) − S(x_{k})]J_{k}(x)e^{i⟨x,P ⟩}v̂(x)dx

+ ∫_{X} S(x)J_{0}(x)e^{i⟨x,P ⟩}v̂(x)dx.

Let v_{k} ∈ S(X^{∗}) be defined by v̂_{k}(x) = J_{k}(x)v̂(x); then the first sum above is ∑_{k=1}^{N} S(x_{k})v_{k}(P). Denote K_{k} = supp J_{k} and let L be the integral of |v̂|. We estimate the last two terms as follows :

∥ ∑_{k=1}^{N} ∫_{X}[S(x) − S(x_{k})]J_{k}(x)e^{i⟨x,P ⟩}v̂(x)dx∥ ≤ sup_{1≤k≤N} sup_{x∈K_{k}} ∥S(x) − S(x_{k})∥L

because ∑_{k=1}^{N} J_{k}(x) ≤ 1, and

∥ ∫_{X} S(x)J_{0}(x)e^{i⟨x,P ⟩}v̂(x)dx∥ ≤ ∥S∥ ∫_{X\K} |v̂(x)|dx.

If K is large enough the second member in the last inequality can be made < ϵ/2. Finally, we have

sup_{x∈K_{k}} ∥S(x) − S(x_{k})∥ ≤ sup_{y∈V} ∥S(y) − S∥

and this can be made < ϵ/(2L) because the function S(·) is continuous. ♢

Corollary 2.2 If A ⊂ C_{bu}(X) is a translation invariant subspace, then [[A · C_{0}(X^{∗})]] = [[C_{0}(X^{∗}) · A]]. In particular, if A is a translation invariant ∗- subalgebra of C_{bu}(X) then [[A · C_{0}(X^{∗})]] is a C^{∗}-subalgebra of B(X).

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

C^{∗}-cross product A ⋊ X is well defined. It can be shown (see [4]) that A ⋊ X is canonically isomorphic to the C^{∗}-subalgebra [[A · C_{0}(X^{∗})]] of B(X). For this reason we shall call this subalgebra cross product of A by X and we use the notation

A ⋊ X = [[A · C_{0}(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 that A ⋊ X coincides with the C^{∗}-algebra generated by the Hamiltonians of the form H = H_{0} + V where H_{0}, the kinetic energy, is fixed, while the interaction is given by a potential V of “type A” in a sense that we shall specify below. So A ⋊ X 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 a C^{∗}-algebra stable under translations and containing the constants A ⊂ C_{bu}(X). We set

A^{∞} = {φ ∈ A ∩ C^{∞}(X) : all the derivatives of φ belong to A}.

We notice that A^{∞} is a dense ∗-subalgebra of A. Indeed, let π ∈ C_{c}^{∞}(X) with ∫ π(x)dx = 1 and let us set θ_{ϵ}(x) = ϵ^{−n}θ(x/ϵ) if ϵ > 0. Then θ_{ϵ} ∗ φ → φ in sup norm as ϵ → 0 if φ ∈ A, because φ is uniformly continuous. So it suffices to prove that all the derivatives of θ_{ϵ} ∗ φ belong to A. Identifying X = ℝ^{n} and taking ϵ = 1, we have for α ∈ ℕ^{n}:

(θ ∗ φ)^{(α)} = θ^{(α)} ∗ φ = ∫ θ^{(α)}(−y)τ_{y}φ dy.

The integral converges in norm in C_{bu}(X) and A is a closed subspace, so (θ ∗ φ)^{(α)} ∈ A.

Theorem 2.3 Let A ⊂ C_{bu}(X) be a translation invariant C^{∗}-subalgebra containing the constant functions and let h : X^{∗} → ℝ be a continuous function such that |h(k)| → ∞ when k → ∞ in X^{∗}. Then the C^{∗}-subalgebra of B(X) generated by the self-adjoint operators of the form h(P + k) + V (Q), where k ∈ X^{∗} and V : X → ℝ belongs to A^{∞}, coincides with A ⋊ X.

Proof. Let C be the C^{∗}-algebra generated by the operators H = h(P + k) + V (Q) ≡ H_{0} + V (Q), with k ∈ X^{∗} and V ∈ A^{∞}. By making a series expansion for large z

(z − H)^{−1} = ∑_{n≥0}(z − H_{0})^{−1}[V (Q)(z − H_{0})^{−1}]^{n}

we easily get C ⊂ A ⋊ X. It remains to prove the opposite inclusion. Let z ∈ ℂ \ h(X^{∗}). Then for μ ∈ ℝ small enough we have z ∉ σ(H_{μ}) if H_{μ} = h(P +k)+μV (Q). The function μ ↦ (H_{μ}−z)^{−1} is norm derivable at μ = 0 with derivative −(H_{0} −z)^{−1}V (Q)(H_{0} −z)^{−1}. Hence (H_{0} −z)^{−1}V (Q)(H_{0} −z)^{−1} ∈ C. Let θ ∈ C_{c}(ℝ) with θ(0) = 1 and ϵ > 0. Then θ(ϵH_{0})(H_{0} − z) = ϵ^{−1}θ_{1}(ϵH_{0}) if θ_{1}(t) = θ(t)(t − z/ϵ). Since ϵH_{0} is affiliated to C, we get θ(ϵH_{0})(H − z) ∈ C. Thus:

θ(ϵH_{0})V (Q)(H_{0} − z)^{−1} = θ(ϵH_{0})(H_{0} − z) · (H_{0} − z)^{−1}V (Q)(H_{0} − z)^{−1} ∈ C.

We can write (H_{0} − z)^{−1} = lim_{n→∞} ψ_{n}(P) (norm limit) with ψ_{n} ∈ C_{c}(X^{∗}). Then, for each s > 0, ψ_{n}(P) is a continuous map from L^{2}(X) into the Sobolev space H^{s}(X) and V (Q) ∈ B(H^{s}(X)). Clearly lim_{ϵ→0} θ(ϵH_{0}) = 1 in norm in B(H^{s}(X), L^{2}(X)). Thus we see that lim_{ϵ→0} θ(ϵH_{0})V (Q)ψ_{n}(P) = V (Q)ψ_{n}(P) in norm in B(L^{2}(X)) for each n. On the other hand, we have ∥V (Q)ψ_{n}(P) − V (Q)(H_{0} −z)^{−1}∥ → 0 as n → ∞. It follows then that lim_{ϵ→0} θ(ϵH_{0})V (Q)(H_{0} − z)^{−1} = V (Q)(H_{0} − z)^{−1} in norm in B(L^{2}(X)).

This argument proves that

V (Q)(h(P + k) − z)^{−1} = V (Q)(H_{0} − z)^{−1} ∈ C

for each k ∈ X^{∗}. This clearly implies V (Q)ξ(H_{0}) ∈ C for ξ ∈ C_{c}(ℝ).

The set of ψ ∈ C_{0}(X^{∗}) such that V (Q)ψ(P) ∈ C is norm closed and contains all the functions of the form ψ(P) = ξ(h(P + k)) with ξ ∈ C_{c}(ℝ) and k ∈ X^{∗}. The family consisting of such functions is a ∗-subalgebra of C_{0}(X^{∗}) which separates the points of X^{∗} (because |h(p + k)| → ∞ if k → ∞). By the Stone-Weierstrass theorem, we see that this family is dense in C_{0}(X^{∗}). So we have V (Q)ψ(P) ∈ C ∀ψ ∈ C_{0}(X^{∗}). Here V is an arbitrary function in A^{∞}. Since A^{∞} is dense in A, we finally obtain φ(Q)ψ(P) ∈ C for all φ ∈ A, ψ ∈ C_{0}(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 in A^{∞}, is equal to A ⋊ X.

Proof. If V ∈ A^{∞}, and if we identify X^{∗} = ℝ^{n}, we have

h(P + k) + V (Q) = h(P) + ∑_{|α|≥1} \frac{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 in C^{∗}-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 infinite L. Note that in the present context it is convenient to interchange the rôles of the lower and upper bounds in the definition of the grading, which explains some differences in notations with respect to [1].

Let L be an arbitrary semilattice, i.e. L is a partially ordered set such that the lower bound a∧b exists for all a, b ∈ L. If L has a least or a greatest element, we denote it min L or max L respectively. We denote by 𝔽(L) the family of finite subsets F ⊂ L such that a ∧ b ∈ F if a, b ∈ F (the empty set belongs to 𝔽(L)). So each F ∈ 𝔽(L) is a finite semilattice for the order relation induced by L, in particular it has a least element min F. We equip 𝔽(L) with the order relation given by inclusion. Since F_{1} ∩F_{2} ∈ 𝔽(L) if F_{1}, F_{2} ∈ 𝔽(L), the set 𝔽(L) becomes a semilattice. Note that for each finite part F ⊂ L the set of elements of the form a_{1} ∧ ⋯ ∧ a_{n} with a_{1}, ⋯, a_{n} ∈ F belongs to 𝔽(L) and contains F.

A L-graded C^{∗}-algebra is a C^{∗}-algebra C equipped with a linearly independent family {C(a)}_{a∈L} of C^{∗}-subalgebras such that:

(i) C(a) · C(b) ⊂ C(a ∧ b) for all a, b ∈ L

(ii) if F ∈ 𝔽(L) then C(F) := ∑_{a∈F} C(a) is a closed subspace of C

(iii) ⋃_{F∈𝔽(L)} C(F) ≡ ∑_{a∈L} C(a) is dense in C.

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-graded C^{∗}-algebra. The set of C^{∗}-subalgebras of C (ordered by inclusion) is a semilattice and F ⟼ C(F) is an injective morphism of semilattices (i.e. C(F_{1} ∩ F_{2}) = C(F_{1}) ∩ C(F_{2})). In particular, we have C(F_{1}) ⊂ C(F_{2}) if F_{1} ⊂ F_{2} and C coincides with the inductive limit of {C(F) : F ∈ 𝔽(L)}, a directed system of C^{∗}-algebras.

We use the notation C(F) = ∑_{a∈F} C(a) for an arbitrary subset F of L. If F is ∧-stable then C(F) is a ∗-algebra, but is not complete in general. In particular C is just the closure of the ∗-algebra C(L). For a ∈ L set L_{a} = {b | a ≤ b}, L′_{a} = {b | a ≰ b} and let C_{a}, J_{a} be the closure of C(L_{a}) and C(L′_{a}) respectively. Note that C(L_{a}) is a ∗-subalgebra of C and C(L′_{a}) is a self-adjoint ideal in C. Hence C_{a}, J_{a} are C^{∗}-subalgebras of C and J_{a} is also a closed self-adjoint ideal in C.

Theorem 3.1 For all a ∈ L one has C = C_{a} + J_{a} and C_{a} ∩ J_{a} = {0}. The projection P_{a} : C → C_{a} determined by this linear direct sum decomposition is a morphism, in particular ∥P_{a}∥ = 1.

Proof. We clearly have C(L) = C(L_{a})+C(L′_{a}) as a linear direct sum. Let P_{a}^{∘} be the projection of C(L) onto C(L_{a}) determined by this decomposition. Thus, if T ∈ C(L) is given by T = ∑_{b∈L} T (b), where T (b) ≠ 0 only for a finite number of b, we have P_{a}^{∘}[T] = ∑_{a≤b} T (b). Then P_{a}^{∘}[T]^{∗} = P_{a}^{∘}[T^{∗}] and if S = ∑_{b∈L} S(b) with S(b) ≠ 0 only for a finite number of b, then

ST = ∑_{b,c} S(b)T (c) = ∑_{d∈L} ∑_{b∧c=d} S(b)T (c).

Thus

P_{a}^{∘}[ST] = ∑_{a≤d} ∑_{b∧c=d} S(b)T (c) = ∑_{a≤b∧c} S(b)T (c)

= ∑_{a≤b, a≤c} S(b)T (c) = P_{a}^{∘}[S] P_{a}^{∘}[T].

Hence P_{a}^{∘} is a morphism of the ∗-algebra C(L) onto its ∗-subalgebra C(L_{a}).

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 the C^{∗}-algebra C(F_{a}), with F_{a} = F ∩ L_{a}. Such a morphism always has norm ≤ 1. Hence we have ∥P_{a}^{∘}[T]∥ ≤ ∥T ∥. Since this is valid for each T ∈ C(L) and C(L) is dense in C, we see that P_{a}^{∘} extends to a morphism P_{a} : C → C_{a} with P_{a}[T] = T if T ∈ C_{a}. In particular, P_{a} is also a linear projection of C onto C_{a} with ∥P_{a}∥ ≤ 1. Since C_{a} ≠ {0} (because C(a) ≠ {0}) we have in fact ∥P_{a}∥ = 1.

We have C(L′_{a}) = kerP_{a}^{∘} ⊂ kerP_{a}. Since C(L) is dense in C, C(L′_{a}) = (1 − P_{a})C(L), and 1 − P_{a} is a continuous surjective map of C onto kerP_{a}, we get that C(L′_{a}) is dense in kerP_{a}. So kerP_{a} = J_{a}. ♢

One can reformulate the preceding theorem in the following terms: the map P_{a}^{∘} : C → C defined by P_{a}^{∘}[∑_{b} T (b)] = ∑_{a≤b} T (b) extends to a norm 1 projection P_{a} of C onto C_{a} which is also a morphism. Not also that the family of C^{∗}- subalgebras {C_{a}}_{a∈L} is decreasing: if a ≤ b then C_{b} ⊂ C_{a} and

P_{a}P_{b} = P_{b}P_{a} = P_{b}  (3.1)

If L has a least element min L then C(min L) is a closed self-adjoint ideal in C, hence one may construct the quotient C^{∗}-algebra

Ĉ := C/C(min L).  (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 ≠ min L such that b ≤ a ⇒ b = min L or b = a. We denote by M the set of atoms of L and we say that L is atomic if each b ≠ min L is minorated by an atom. Then we can associate to C a second C^{∗}-algebra, namely

C̃ := ⨁_{a∈M} C_{a}  (3.3)

where the direct sum is in the C^{∗}-algebra sense. Observe that there is a natural morphism P : C → C̃, namely

P[T] = (P_{a}[T])_{a∈M}.

Theorem 3.2 Assume that the semilattice L has a least element and is atomic. Then the kernel of the morphism P is equal to C(min L).

Proof. First we note that the result is known (and easy to prove) if L is finite, see Theorem 8.4.1 in [1]; this particular case will be needed below. Clearly P_{a}[T] = 0 if T ∈ C(min L) and a ≠ min L, so C(min L) ⊂ Ker P. Reciprocally, let T ∈ C such that P_{a}[T] = 0 for all a ∈ M. Then for each ϵ > 0 there is F ∈ 𝔽(L) and there is S ∈ C(F) such that ∥T − S∥ ≤ ϵ. We assume, without loss of generality, that min L ∈ F, hence F is a finite semilattice with min F = min L. If b ≠ min L is an element of F then there is a ∈ M such that a ≤ b, hence P_{b}[T] = P_{b}P_{a}[T] = 0. We get

∥P_{b}[S]∥ = ∥P_{b}[T − S]∥ ≤ ∥T − S∥ ≤ ϵ.

Here P_{b} is the projection associated to the algebra C. However, it is clear (see the remark after the proof of Theorem 3.1) that the restriction of P_{b} to C(F) coincides with the canonical projection of the F-graded algebra C(F) onto its subalgebra C(F_{b}), F_{b} = {c ∈ F : b ≤ c}.

Let N be the set of atoms of F. Then according to Theorem 8.4.1 from [1], the map U ↦ (P_{b}[U])_{b∈N}, sending C(F) into ⨁_{b∈N} C(F_{b}), has C(min F) = C(min L) as its kernel. The map C(F)/C(min L) → ⨁_{b∈N} C(F_{b}) will be an isometry and since ∥P_{b}[S]∥ ≤ ϵ for each b ∈ N, the image of S in the quotient space C(F)/C(min L) has norm ≤ ϵ. From the definition of the quotient norm it follows that there is S_{0} ∈ C(min L) such that ∥S − S_{0}∥ ≤ 2ϵ (in fact ≤ ϵ).

Thus, we see that for each ϵ > 0 there is S_{0} ∈ C(min L) such that

∥T − S_{0}∥ = ∥T − S + S − S_{0}∥ ≤ 3ϵ.

Since C(min L) is closed we get T ∈ C(min L). ♢

The preceding theorem gives us a canonical embedding Ĉ ⊂ C̃, more precisely

C/C(min L) ↪ ⨁_{a∈M} C_{a}.  (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 and Y a linear subspace. We denote π_{Y} = π_{Y}^{X} the canonical surjection of X onto the quotient vector space X/Y and Y^{⊥} the set of x^{∗} ∈ X^{∗} such that ⟨y, x^{∗}⟩ = 0 ∀y ∈ Y. We have canonical identifications (X/Y)^{∗} = Y^{⊥} and X^{∗}/Y^{⊥} = Y^{∗}.

We shall embed C_{0}(X/Y) ⊂ C_{b}(X) with the help of the map φ ⟼ φ ∘ π_{Y}. Since C_{b}(X) ⊂ B(X), we shall have

C_{0}(X/Y) ⊂ C_{bu}(X/Y) ⊂ C_{bu}(X) ⊂ B(X).  (4.1)

For φ ∈ C_{b}(X/Y) we shall denote φ(Q_{Y}) = (φ ∘ π_{Y})(Q) the operator in B(X) associated to it. Sometimes it is important to specify in the notations the space X; then we set φ(Q_{Y}) = φ(Q_{Y}^{X}).

The relation Y^{∗} = X^{∗}/Y^{⊥} implies

C_{b}(Y^{∗}) = C_{b}(X^{∗}/Y^{⊥}) ⊂ C_{b}(X^{∗}) ⊂ B(X).  (4.2)

For ψ ∈ C_{b}(Y^{∗}) we denote ψ(P_{Y}) or ψ(P_{Y}^{X}) the operator in B(X) associated to it; we have ψ(P_{Y}^{X}) = Φ_{X^{∗}}[ψ(Q_{Y^{⊥}}^{X^{∗}}]. Observe, in particular, that the group C^{∗}-algebra C_{0}(Y^{∗}) of the additive group Y is embedded in B(X).

Let 𝔾(X) be the Grassmannian of X, 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 ∩Z and Y ∨Z = Y +Z. For each Y ∈ 𝔾(X) we have a C^{∗}-subalgebra C_{0}(X/Y) of C_{b}(X) as explained above. In particular C_{0}(X/O) = C_{0}(X) and C_{0}(X/X) = ℂ. Note that each C_{0}(X/Y) is translation invariant, i.e. it is stable under all the automorphisms τ_{x}, x ∈ X.

If F ⊂ 𝔾(X) is a family of vector subspaces of X then we set

C_{0}^{X}(F) = ∑_{Y ∈F} C_{0}(X/Y).  (4.3)

This is the linear subspace of C_{bu}(X) generated by ⋃_{Y ∈F} C_{0}(X/Y). Note that C_{0}^{X}(∅) = {0} and C_{0}^{X}(Y) ≡ C_{0}^{X}({Y }) = C_{0}(X/Y).

Lemma 4.1 (a) The family {C_{0}(X/Y) : Y ∈ 𝔾(X)} of C^{∗}-subalgebras of C_{b}(X) is linearly independent.

(b) If F ⊂ 𝔾(X) is finite then C_{0}^{X}(F) is a closed subspace of C_{bu}(X).

(c) For each Y, Z ∈ 𝔾(X) the set C_{0}(X/Y) · C_{0}(X/Z) is a dense subalgebra of C_{0}(X/(Y ∩ Z)).

Proof. We give a detailed proof of this simple lemma because the same argument will be used later on in order to prove Theorem 4.5. Let F ⊂ 𝔾(X) be finite and for each Y ∈ F let φ_{Y} ∈ C_{0}(X/Y). Denote φ^{Y} : X/Y → ℂ the function such that φ_{Y} = φ^{Y} ∘ π_{Y}. Then for each ω ∈ X one has (τ_{ω}φ_{Y})(x) = φ^{Y} (π_{Y} (x)+π_{Y} (ω)) for all x ∈ X. Hence, if we set F_{ω} = {Y ∈ F : ω ∈ Y }, then

lim_{λ→∞} τ_{λω}[ ∑_{Y ∈F} φ_{Y}] = ∑_{Y ∈F_{ω}} φ_{Y}  (4.4)

pointwise on X. In particular

∥ ∑_{Y ∈F_{ω}} φ_{Y} ∥ ≤ ∥ ∑_{Y ∈F} φ_{Y} ∥  (4.5)

where ∥ · ∥ is the sup norm.

Let us prove that there is a number C such that for all Z ∈ F and all {φ_{Y} } as above

∥φ_{Z}∥ ≤ C∥ ∑_{Y ∈F} φ_{Y} ∥.  (4.6)

This clearly implies (a) and (b). If the set

Z_{0} = Z \ ⋃_{Y ≠Z} Y = ⋂_{Y ≠Z}[Z \ (Y ⋂ Z)]

is not empty then (4.5) with a choice ω ∈ Z_{0} gives (4.6) (with C = 1). Since Y ∩ Z are linear subspaces of Z one has Z_{0} = ∅ if and only if there is Y ∈ F such that Z ⊂ Y strictly. This cannot happen if Z is a maximal element in F, hence (4.6) holds for such elements. Let F_{1} be the set of Y ∈ F which are not maximal elements in F. Then we clearly get ∥ ∑_{Y ∈F_{1}} φ_{Y} ∥ ≤ C_{1}∥ ∑_{Y ∈F} φ_{Y} ∥ for some constant C_{1}. By what we already proved we see then that (4.6) holds for the maximal elements Z of F_{1}, etc.

We now prove (c). Let E = (X/Y) × (X/Z) equipped with the direct sum vector space structure. If φ ∈ C_{0}(X/Y) and ψ ∈ C_{0}(X/Z) then φ ⊗ ψ denotes the function (s, t) ⟼ φ(s)ψ(t), which belongs to C_{0}(E). The subspace generated by the functions of the form φ ⊗ ψ is dense in C_{0}(E) by the Stone-Weierstrass theorem. Let F be a linear subspace of E. Since each function in C_{0}(F) extends to a function in C_{0}(E) we see that the restrictions (φ ⊗ ψ) | F generate a dense linear subspace of C_{0}(F).

Let us denote by π the map x ⟼ (π_{Y} (x), π_{Z}(x)), so π is 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 π̃ : X/V → F such that π = π̃ ∘ π_{V}. So θ ⟼ θ ∘ π̃ is an isometric isomorphism of C_{0}(F) onto C_{0}(X/V). Hence for φ ∈ C_{0}(X/Y) and ψ ∈ C_{0}(X/Z) the function θ = (φ ⊗ ψ) ∘ π̃ belongs to C_{0}(X/V), it has the property θ ∘π_{V} = φ∘π_{Y} ·ψ ∘π_{Z}, and the functions of this form generate a dense linear subspace of C_{0}(X/V). ♢

We say that F ⊂ 𝔾(X) is ∩-stable if Y, Z ∈ F ⇒ Y ∩ Z ∈ F (so F is a generalized flag of subspaces of X). Such a F 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 of 𝔾(X).

Corollary 4.2 If F ∈ 𝔽(X) then C_{0}^{X}(F) is a C^{∗}-subalgebra of C_{bu}(X) equipped with a natural structure of F-graded C^{∗}-algebra. This algebra is unital if and only if X ∈ F. For F_{1}, F_{2} ∈ 𝔽(X) one has

C_{0}^{X}(F_{1}) ⋂ C_{0}^{X}(F_{2}) = C_{0}^{X}(F_{1} ⋂ F_{2}).  (4.7)

In particular, one has C_{0}^{X}(F_{1}) ⊂ C_{0}^{X}(F_{2}) if and only if F_{1} ⊂ F_{2}.

4.2

We are ready to define the noncommutative versions of the algebras C_{0}(X/Y): they are cross products of algebras of the form C_{0}(X/Y) by the natural action τ of the additive group X. 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 C_{0}^{X}(Y) = C_{0}(X/Y) ⋊ X is the C^{∗}-subalgebra of B(X) obtained as norm closure of C_{0}(X/Y) · C_{0}(X^{∗}). For each subset F ⊂ 𝔾(X) let C_{0}^{X}(F) be the linear subspace of B(X) generated by the algebras C_{0}^{X}(Y) with Y ∈ F, so

C_{0}^{X}(F) = ∑_{Y ∈F} C_{0}^{X}(Y).  (4.8)

Observe that C_{0}^{X}(F) = [[C_{0}^{X}(F) · C_{0}(X^{∗})]] = [[C_{0}(X^{∗}) · C_{0}^{X}(F)]]. So for ∩-stable F we have C_{0}^{X}(F) = C_{0}^{X}(F) ⋊ X.

To each vector subspace Y of X we have thus associated a C^{∗}-subalgebra C_{0}^{X}(Y) of B(X). The only one which is abelian is

C_{0}^{X}(X) = C_{0}(X^{∗}) = {φ(P) | φ ∈ C_{0}(X^{∗})}.  (4.9)

The algebra C_{0}^{X}(O) is generated by C_{0}(X) · C_{0}(X^{∗}) and, since the operators of the form φ(Q)ψ(P)) with φ ∈ C_{0}(X), ψ ∈ C_{0}(X^{∗}) are compact, we have

C_{0}^{X}(O) = K(X).  (4.10)

The algebras which play the main rôle in the N-body problem (as presented in ch. 9 of [1]) are of the form C_{0}^{X}(F) with finite F and will be studied in this section. The next one is devoted to the case F = 𝔾(X).

We shall need an extension of the automorphism τ_{x} of C_{b}(X) to an automorphism of B(X): we set τ_{x}[S] = e^{i⟨x,P ⟩}Se^{−i⟨x,P ⟩} for each x ∈ X and S ∈ B(X). Observe that for φ ∈ C_{0}(X/Y) and ψ ∈ C_{0}(X^{∗}) one has

τ_{x}[φ(Q_{Y})ψ(P)] = φ(Q_{Y} + π_{Y} (x))ψ(P).

This immediately gives the next lemma.

Lemma 4.4 (i) If y ∈ Y and S ∈ C_{0}^{X}(Y) then τ_{y}[S] = S; (ii) if S ∈ C_{0}^{X}(Y) and π_{Y} (x) → ∞ then τ_{x}[S] → 0 in the strong operator topology.

Theorem 4.5 (a) The family {C_{0}^{X}(Y) : Y ∈ 𝔾(X)} of C^{∗}-subalgebras of B(X) is linearly independent.

(b) If F ⊂ 𝔾(X) is finite then C_{0}^{X}(F) is (norm) closed in B(X).

(c) If Y, Z ∈ 𝔾(X) then C_{0}^{X}(Y) · C_{0}^{X}(Z) is a dense linear subspace of the C^{∗}- algebra C_{0}^{X}(Y ∩ Z).

Proof. Let F ⊂ 𝔾(X) be finite and for each Y ∈ F let T (Y) ∈ C_{0}^{X}(Y). Then

s − lim_{λ→∞} τ_{λω}[ ∑_{Y ∈F} T (Y)] = ∑_{Y ∈F_{ω}} T (Y)  (4.11)

where the notations are as in the proof of Lemma 4.1. Indeed, this is an immediate 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 2.2 C_{0}^{X}(Y) is the norm closed linear space generated by the operators of the form ψ_{Y} (P)φ_{Y} (Q_{Y}), with ψ_{Y} ∈ C_{0}(X^{∗}) and φ_{Y} ∈ C_{0}(X/Y). On the other hand C_{0}^{X}(Z) is, by definition, the norm closure of the linear space generated by the operators of the form φ_{Z}(Q_{Z})ψ_{Z}(P), with φ_{Z} ∈ C_{0}(X/Z) and ψ_{Z} ∈ C_{0}(X^{∗}). By Lemma 4.1 one has φ_{Y} (Q_{Y})φ_{Z}(Q_{Z}) = φ_{V} (Q_{V}) for some φ_{V} ∈ C_{0}(X/V), where V = Y ∩ Z. So

ψ_{Y} (P)φ_{Y} (Q_{Y}) · φ_{Z}(Q_{Z})ψ_{Z}(P) = ψ_{Y} (P) · φ_{V} (Q_{V})ψ_{Z}(P)

which clearly belongs to C_{0}^{X}(Y ∩ Z). This proves that C_{0}^{X}(Y) · C_{0}^{X}(Z) ⊂ C_{0}^{X}(Y ∩ Z).

Elements of the form ψ_{1}(P)φ_{V} (Q_{V})ψ_{2}(P), with ψ_{1}, ψ_{2} ∈ C_{0}(X^{∗}) and φ_{V} ∈ C_{0}(X/V), clearly generate C_{0}^{X}(Y ∩ Z) (because those elements of the form φ_{V} (Q_{V})ψ_{1}(P)ψ_{2}(P) do and we may use Lemma 2.1). Hence the density of C_{0}^{X}(Y) · C_{0}^{X}(Z) in C_{0}^{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 C_{0}^{X}(F) is a C^{∗}-subalgebra of B(X). If we equip F with the order relation given by inclusion then the family {C_{0}^{X}(Y)}_{Y ∈F} of C^{∗}-subalgebras of C_{0}^{X}(F) provides C_{0}^{X}(F) with a structure of F-graded C^{∗}- algebra.

4.3

The choice of a supplementary subspace Z of Y in X will give us a canonical isomorphism between C(X/Y) and the C^{∗}-tensor product of the algebras C_{0}(Y^{∗}) and K(Z):

C(X/Y) ≅ C_{0}(Y^{∗}) ⊗ K(Z).  (4.12)

In order to define in a precise way this isomorphism let us introduce some notations. Since X = Y + Z (direct sum) we have a canonical identification of X^{∗} with Y^{∗} ⊕ Z^{∗}. Let i_{Y} , i_{Z} be the inclusion maps of Y, Z into X respectively. By taking adjoints we get an isomorphism (i_{Y}^{∗} , i_{Z}^{∗}) : X^{∗} → Y^{∗} ⊕ Z^{∗} and we may define for any functions u : Y^{∗} → ℂ and v : Z^{∗} → ℂ the function u⊗v : X^{∗} → ℂ by (u ⊗ v)(x^{∗}) = u(i_{Y}^{∗} x^{∗})v(i_{Z}^{∗}x^{∗}). If u ∈ C_{0}(Y^{∗}), v ∈ C_{0}(Z^{∗}) then clearly u⊗v ∈ C_{0}(X^{∗}). Moreover, the projection p_{Z} : X → Z determined by the direct sum decomposition X = Y +Z factorizes to an isomorphism p_{Z}^{♭} : X/Y → Z and if for w : Z → ℂ we define w^{♭} : X/Y → ℂ by w^{♭} = w ∘ p_{Z}^{♭}, then w^{♭} ∈ C_{0}(X/Y) if w ∈ C_{0}(Z).

Proposition 4.7 There is a linear continuous map C_{0}(Y^{∗}) ⊗ K(Z) → C_{0}^{X}(Y) such that for each u ∈ C_{0}(Y^{∗}), v ∈ C_{0}(Z^{∗}) and w ∈ C_{0}(Z) the element u(P^{Y})⊗ [w(Q^{Z})v(P^{Z})] is sent into w^{♭}(Q_{Y})(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 form u(P^{Y}) ⊗ [w(Q^{Z})v(P^{Z})] and w^{♭}(Q_{Y})(u ⊗ v)(P^{X}) span dense linear subspaces in C_{0}(Y^{∗}) ⊗ K(Z) and C_{0}^{X}(Y) respectively. To prove the existence and the isomorphism properties observe that we get an isomorphism J : H(X) → H(Y) ⊗ H(Z) by setting (J f)(y, z) = f(y + z). It remains to check that

u(P^{Y}) ⊗ [w(Q^{Z})v(P^{Z})] · J = J · w^{♭}(Q_{Y})(u ⊗ v)(P^{X})

which is a straightforward consequence of the definitions. ♢

The preceding tensor product decomposition of H(X) also gives canonical isomorphisms C_{0}(X/Y) ≅ 1 ⊗ C_{0}(Z) and C_{0}(X^{∗}) ≅ C_{0}(Y^{∗}) ⊗ C_{0}(Z^{∗}). This induces a linear isomorphism of the vector spaces C_{0}(X/Y) · C_{0}(X^{∗}) and C_{0}(Y^{∗}) ⊗ [C_{0}(Z) · C_{0}(Z^{∗})] which extends to the isomorphism between C_{0}^{X}(Y) and C_{0}(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 C_{0}^{X}(Y) will be equal to C_{0}(Y^{∗}) ⊗ K(Y_{α}^{⊥}). The algebras C_{0}(Y^{∗}) and C_{0}(Y^{∗}) ⊗ K(Y_{α}^{⊥}) are denoted 𝕋(Y) and T (Y_{α}^{⊥}) in [1]. If α is replaced by a new scalar product β, so that Y has a different orthogonal subspace Y_{β}^{⊥}, then C_{0}(Y^{∗}) ⊗ K(Y_{β}^{⊥}) gives (after the identification H(X) = H(Y) ⊗ H(Y_{β}^{⊥})) the same algebra as C_{0}(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 C_{0}^{X}(Y) stresses this fact.

5 The Algebra ℭ_{0}^{X}

5.1

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

C_{0}^{X} := norm closure in B(X) of C_{0}^{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) = C_{0}^{X}(Y) for Y ∈ 𝔾(X). So the algebra C_{0}^{X} is 𝔾(X)-graded and can be identified with the inductive limit of the family of C^{∗}-algebras {C_{0}^{X}(F) | F ∈ 𝔽(X)}. Indeed, if we order 𝔽(X) by the inclusion relation then C_{0}^{X}(F_{1}) ⊂ C_{0}^{X}(F_{2}) if and only if F_{1} ⊂ F_{2} and

⋃_{F∈𝔽(X)} C_{0}^{X}(F) = C_{0}^{X}(𝔾(X))  (5.2)

is a dense ∗-subalgebra of C_{0}^{X}. Moreover, for all F_{1}, F_{2} in 𝔽(X) we have

C_{0}^{X}(F_{1}) ∩ C_{0}^{X}(F_{2}) = C_{0}^{X}(F_{1} ∩ F_{2}).  (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 spaces ∑_{Y ⊂Z} C_{0}^{X}(Z) and ∑_{Y ⊄Z} C_{0}^{X}(Z) respectively. Observe that the notations are consistent: if Y = O = {0} then C_{O}^{X} = C_{0}^{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) of C_{0}^{X} onto its subspace C_{Y}^{X}.

Theorem 5.1 C_{Y}^{X} is a C^{∗}-subalgebra of C_{0}^{X}, J_{Y}^{X} is a closed self-adjoint ideal in C_{0}^{X}, and C_{0}^{X} is equal to their direct sum: C_{0}^{X} = C_{Y}^{X} + J_{Y}^{X} and C_{Y}^{X} ⋂ J_{Y}^{X} = {0}. The linear projection P_{Y} of the linear space C_{0}^{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)} of C^{∗}-subalgebras of C_{0}^{X} is decreasing

Y ⊂ Z ⇒ C_{Z}^{X} ⊂ C_{Y}^{X}  (5.4)

and has a least element C_{X}^{X} = C_{0}(X^{∗}). Clearly (5.4) implies

P_{Z}P_{Y} = P_{Y} P_{Z} = P_{Z}.  (5.5)

Our purpose now is to describe the quotient of the algebra C_{0}^{X} with respect to the ideal C_{0}^{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 C̃_{0}^{X} the C^{∗}-direct sum of the algebras C_{Y}^{X} with Y ∈ ℙ(X):

C̃_{0}^{X} = ⨁_{Y ∈ℙ(X)} C_{Y}^{X}.  (5.6)

Let P : C_{0}^{X} → C̃_{0}^{X} be defined by

P[T] = ⨁_{Y ∈ℙ(X)} P_{Y} [T].  (5.7)

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

So P induces an embedding of the C^{∗}-algebra C_{0}^{X}/K(X) into C̃_{0}^{X}. We shall identify C_{0}^{X}/K(X) with a subalgebra of C̃_{0}^{X}:

C_{0}^{X}/K(X) ⊂ C̃_{0}^{X}.  (5.8)

5.2

We shall make here some final remarks concerning the algebra C_{0}^{X}. First we give another description of the maps P_{Y}. Observe that by Theorem 4.5(a) each T ∈ C_{0}^{X}(𝔾(X)) can be written in a unique way as a sum T = ∑_{Z∈F} T (Z) with F ⊂ 𝔾(X) finite and T (Z) ∈ C_{0}^{X}(Z), T (Z) ≠ 0. For such a T we have

P_{Y} [T] = ∑_{Z∈F,Z⊃Y} T (Z)  (5.9)

and this property uniquely characterizes P_{Y}. If ω ∈ Y is such that ω ∉ Z if Z ∈ F and Y ⊄ Z (such a choice is possible because F is finite), then

s − lim_{λ→∞} τ_{λω}[T] = ∑_{Z⊃Y} T (Z),

by Lemma 4.4. In other terms, for T as above we have

P_{Y} [T] = s − lim_{λ→∞} τ_{λω}[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 all T ∈ C_{0}^{X}:

P_{Y} [T] = s − lim_{λ→∞} τ_{λω}[T].  (5.11)

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

Notice that there is an abelian version of the algebra C_{0}^{X}, namely the closure C_{0}^{X} in C_{b}(X) of ∑_{Y ⊂X} C_{0}(X/Y), and everything we have done applies to C_{0}^{X} too. In particular, for f ∈ C_{0}^{X} we have: f ∈ C_{0}(X) if and only if lim_{λ→∞} f(x + λω) = 0 for each ω ∈ X \ {0}. A geometric proof of this not obvious fact (if dim X > 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 of N-body Hamiltonians. We shall briefly present them and their relation with the algebras C_{0}^{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 C_{0}^{X} is generated by functions of the form φ(Q)ψ(P) (or ψ(P)φ(Q)) with ψ ∈ S(X^{∗}), it is easy to prove that [χ(Q), T] ∈ K(X) for all T ∈ C_{0}^{X}.

Now let Z ⊂ X be a subspace and assume that χ(z) = 0 if z ∈ Z, |z| ≥ 1. Then for each φ ∈ C_{0}(X/Z) the function χ · φ ∘ π_{Z} belongs to C_{0}(X) (indeed, if x → ∞ and π_{Z}(x) is bounded, then x/|x| approaches Z), hence χ(Q)φ(Q_{Z})ψ(P) is a compact operator. It follows that χ(Q)T and T χ(Q) are compact operators if T ∈ C_{0}^{X}(Z).

Let us fix F ∈ 𝔽(X) with O, X ∈ F and let Y ∈ F, Y ≠ X. A C^{∞} function χ_{Y} : X → ℝ which is homogeneous of degree zero on |x| ≥ 1 is called (according to Froese-Herbst) Y -reducing if: for each Z ∈ F with Y ⊄ Z and each z ∈ Z, |z| ≥ 1, one has χ_{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)T and T χ_{Y} (Q) belong to K(X) if T ∈ C(F′_{Y}) = ∑_{Z∈F,Y ⊄Z} C_{0}^{X}(Z).

Let N be the set of atoms of F. An F-reducing partition of unity on X is a family {χ_{Y} }_{Y ∈N} such that χ_{Y} is Y -reducing and ∑_{Y ∈N} χ_{Y}^{2} = 1 on X. In [3] such families are constructed. From (i) and (ii) above we then get: if S ∈ C(F) and if we denote S_{Y} = P_{Y} [S] its canonical projection onto C(F_{Y}), then there is K ∈ K(X) such that

S = K + ∑_{Y ∈N} χ_{Y} (Q) S_{Y} χ_{Y} (Q).  (5.12)

It is clear that F cannot be replaced by an infinite semilattice in the preceding construction. However, these partitions can be used to give an alternate and more elementary proof of the main assertion of Theorem 5.2, namely that T is compact if T ∈ C_{0}^{X} and w − lim_{λ→∞} τ_{λω}[T] = 0 for each ω ∈ X \ {0}. Indeed, for each ϵ > 0 we can find F as above and S ∈ C(F) such that ∥T − S∥ ≤ ϵ. Note that we can assume T and S self-adjoint. Write S = ∑{S(Z) : Z ∈ F} with S(Z) ∈ C_{0}^{X}(Z), hence S_{Y} = ∑{S(Z) : Z ∈ F, Z ⊃ Y }, and let F_{ω} = {Z ∈ F : ω ∈ Z} for ω ∈ X \ {0}. Then s − lim_{λ→∞} τ_{λω}[S] = ∑_{Z∈F_{ω}} S(Z) and from the Fatou lemma we get ∥ ∑_{Z∈F_{ω}} S(Z)∥ ≤ ϵ. For each Y ∈ N we can find ω ∈ Y \ {0} such that ω ∉ Y′ if Y′ ∈ N \ {Y }. Hence we get ∥S_{Y} ∥ ≤ ϵ, or −ϵ ≤ S_{Y} ≤ ϵ, for each Y ∈ N. Then −ϵ ≤ ∑ χ_{Y} (Q) S_{Y} χ_{Y} (Q) ≤ ϵ because ∑ χ_{Y} (Q)^{2} = 1. So from (5.12) we see that there is K ∈ K(X) such that ∥S − K∥ < ϵ. This implies ∥T − K∥ ≤ 2ϵ, which proves the assertion.

References

[1] W. Amrein, A. Boutet de Monvel, and V. Georgescu, C_{0}-Groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians, Birkhäuser, Progress in Math. Ser., 135, 1996.

[2] M. Damak and V. Georgescu, C^{∗}-Algebras Related to the N-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 of N-Body Schrödinger 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: Vladimir.Georgescu@math.pst.u-cergy.fr