Fuzzy hyperspheres via confining potentials and energy cutoffs

We simplify and complete the construction of fully $O(D)$-equivariant fuzzy spheres $S^d_L$, for all dimensions $d\equiv D-1$, initiated in [G. Fiore, F. Pisacane, J. Geom. Phys. 132 (2018), 423]. This is based on imposing a suitable energy cutoff on a quantum particle in $\mathbb{R}^D$ in a confining potential well $V(r)$ with a very sharp minimum on the sphere of radius $r=1$; the cutoff and the depth of the well diverge with $L\in\mathbb{N}$. As a result, the noncommutative Cartesian coordinates $\overline{x}^i$ generate the whole algebra of observables $A_L$ on the Hilbert space $H_L$; $H_L$ can be recovered applying polynomials in the $\overline{x}^i$ to any of its elements. The commutators of the $\overline{x}^i$ are proportional to the angular momentum components, as in Snyder noncommutative spaces. $H_L$, as carrier space of a reducible representation of $O(D)$, is isomorphic to the space of harmonic homogeneous polynomials of degree $L$ in the Cartesian coordinates of (commutative) $\mathbb{R}^{D+1}$, which carries an irreducible representation ${\bf\pi}_L$ of $O(D+1)\supset O(D)$. Moreover, $A_L$ is isomorphic to ${\bf\pi}_L\left(Uso(D+1)\right)$. We resp. interpret $\{H_L\}_{L\in\mathbb{N}}$, $\{A_L\}_{L\in\mathbb{N}}$ as fuzzy deformations of the space $H_s:={\cal L}^2(S^d)$ of (square integrable) functions on $S^d$ and of the associated algebra $A_s$ of observables, because they resp. go to $H_s,A_s$ as $L$ diverges (with $\hbar$ fixed). With suitable $\hbar=\hbar(L)\stackrel{L\to\infty}{\longrightarrow} 0$, in the same limit $A_L$ goes to the (algebra of functions on the) Poisson manifold $T^*S^d$; more formally, $\{A_L\}_{L\in\mathbb{N}}$ yields a fuzzy quantization of a coadjoint orbit of $O(D+1)$ that goes to the classical phase space $T^*S^d$.


Introduction
Noncommutative space(time) algebras are introduced and studied with various motivations, notably to provide an arena for regularizing ultraviolet (UV) divergences in quantum field theory (QFT) (see e.g. [1]), reconciling Quantum Mechanics and General Relativity in a satisfactory Quantum Gravity (QG) theory (see e.g. [2]), unifying fundamental interactions (see e.g. [3,4]). Noncommutative Geometry (NCG) [5,8,6,7] has become a sophisticated framework that develops the whole machinery of differential geometry on noncommutative spaces. Fuzzy spaces are particularly appealing noncommutative spaces: a fuzzy space is a sequence {A} n∈N of finite-dimensional algebras such that A n n→∞ −→ A ≡algebra of regular functions on an ordinary manifold, with dim(A n ) n→∞ −→ ∞. They have raised a big interest in the high energy physics community as a non-perturbative technique in QFT based on a finite discretization of space(time) alternative to the lattice one: the main advantage is that the algebras A n can carry representations of Lie groups (not only of discrete ones). They can be used also for internal (e.g. gauge) degrees of freedom (see e.g. [9]), or as a new tool in string and D-brane theories (see e.g. [10,11]). The first and seminal fuzzy space is the 2-dimensional Fuzzy Sphere (FS) of Madore and Hoppe [12,13], where A n M n (C), which is generated by coordinates x i (i = 1, 2, 3) fulfilling (sum over repeated indices is understood); they are obtained by the rescaling x i = 2L i / √ n 2 −1 of the elements L i of the standard basis of so (3) in the unitary irreducible representation (irrep) (π l , V l ) of dimension n = 2l+1, i.e. where V l is the eigenspace of the Casimir L 2 = L i L i with eigenvalue l(l + 1). Ref. [14,15] first proposed a QFT based on it. Each matrix in M n can be expressed as a polynomial in the x i that can be rearranged as the expansion in spherical harmonics of an element of C(S 2 ) truncated at level n. Unfortunately, such a nice feature is not shared by the fuzzy spheres of dimension d > 2: the product of two spherical harmonics is not a combination of spherical harmonics, but an element in a larger algebra A n . Fuzzy spheres of dimension d = 4 and any d ≥ 3 were first introduced respectively in [16] and [17]; other versions in d = 3, 4 or d ≥ 3 have been proposed in [18,19,20,21].
The Hilbert space of a (zero-spin) quantum particle on configuration space S d and the space of continuous functions on S d carry (the same) reducible representation of O(D), with D := d+1; this decomposes into irreducible representations (irreps) as follows where the carrier space V l D is an eigenspace of the quadratic Casimir L 2 with eigenvalue E l := l(l+D−2) (V l 3 ≡ V l ). C(S d ) can be seen as an algebra of bounded operators on L 2 (S d ). On the contrary, the mentioned fuzzy hyperspheres (including the Madore-Hoppe FS) are either based on sequences of irreps of Spin(D) (so that r 2 , which is proportional to L 2 , is identically 1) parametrized by n [12,13,16,17,18,19], or on sequences of reducible representations that are the direct sums of small bunches of such irreps [20,21]. In either case, even excluding the n for which the associated representation of O(D) is only projective, the carrier space does not go to (2) in the limit n → ∞; we think this makes the interpretation of these fuzzy spheres as fuzzy configuration spaces S d (and of the x i as spatial coordinates) questionable. For the Madore-Hoppe FS such an interpretation is even more difficult, because relations (1) are equivariant under SO(3), but not under the whole O(3), e.g. not under parity x i → −x i , while the ordinary sphere S 2 is; on the contrary, all the other mentioned fuzzy spheres are O(D)-equivariant, because the commutators [x i , x j ] are Snyder-like [1], i.e. proportional to angular momentum components L ij .
The purpose of this work is to complete the construction [22,23] of new, fully O(D)equivariant fuzzy quantizations of spheres S d of arbitrary dimension d = D − 1 ∈ N (thought as configuration spaces) and of T * S d (thought as phase spaces), in a sense that will be fully clarified at the end of section 7; in the commutative limit the involved O(D)-representation goes to (2). We also simplify and uniformize (with respect to D) the procedure of [22,23].
We recall this procedure starting from the general underlying philosophy [24,22,25]. Consider a quantum theory T with Hilbert space H, algebra of observables on H (or with a domain dense in H) A ≡ Lin(H), Hamiltonian H ∈ A. For any subspace H ⊂ H preserved by the action of H, let P : H → H be the associated projector and the observable A ≡ P AP ∈ A will have the same physical interpretation as A. By construction H = P H = HP . The projected Hilbert space H, algebra of observables A and Hamiltonian H provide a new quantum theory T . If H, H are invariant under some group G, then P , A, H, T will be as well. In general, the relations among the generators of A differ from those among the generators of A. In particular, if the theory T is based on commuting coordinates x i (commutative space) this will be in general no longer true for T : [x i , x j ] = 0, and we have generated a quantum theory on a NC space.
A physically relevant instance of the above projection mechanism occurs when H is the subspace of H characterized by energies E below a certain cutoff, E ≤ E; then T is a low-energy effective approximation of T . What it can be useful for? If A contains all the observables corresponding to measurements that we can really perform with the experimental apparati at our disposal, and the initial state of the system belongs to H, then neither the dynamical evolution ruled by H, nor any measurement can map it out of H, and we can replace T by the effective theory T . Moreover, if at E > E we even expect new physics not accountable by T , then T may also help to figure out a new theory T valid for all E.
For an ordinary (for simplicity, zero-spin) quantum particle in the Euclidean (configuration) space R D it is H = L 2 (R d ). Fixed a Hamiltonian H(x, p), by standard wisdom the dimension of H is where h is the Planck constant and B E ≡ (x,p) ∈ R 2D | H(x,p) ≤ E is the classical phase space region with energy below E. If H consists only of the kinetic energy T , then this is infinite ( fig. 1 left). If H = T +V , with a confining potential V , then this is finite ( fig. 1 right) at least for sufficiently small E, and also the classical region v E ⊂ R D in configuration space determined by the condition V ≤ E is bounded. In the sequel we rescale x, p, H, V so that they are dimensionless and, denoting by ∆ the Laplacian in R D , we can write The 'dimensional reduction' R D S d of the configuration space is obtained:  3. Making both k, E depend on, grow and diverge with a natural number Λ. Thereby we rename H, P , A as H Λ , P Λ , A Λ .
As H is O(D)-invariant, so are P Λ , H = P Λ H, and the projected theory is O(D)-equivariant.
Technical details are given in sections 2, 3. Section 2 fixes the notation and contains preliminaries partly developed in [23]. The representation-theoretical results of section 3, which deserve attention also on their own, allow to explictly characterize the space V l D as the space of harmonic homogeneous polynomials of degree l in the Cartesian coordinates x i of R D restricted to the sphere S d ; we determine such polynomials constructing the trace-free completely symmetric projector of R D ⊗ l and applying it to the homogeneous polynomials of degree l in x i . The actions of the L hk and of the multiplication operators x i · on such polynomials can be expressed by general formulae valid for all D, l; this allows to avoid the rather complicated actions of the L hk on spherical harmonics (which also span V l D ) used in [23]. It turns out that both H Λ , V Λ D+1 decompose into irreps of O(D) as follows: The second equality shows that, in contrast with the mentioned fuzzy hyperspheres, we recover (2) in the limit Λ → ∞. The first equality suggests that the unitary irrep of the * -algebra A Λ on H Λ is isomorphic to the irrep π Λ of U so(D + 1) on V Λ D+1 , what we in fact prove in section 5 (this had been proved for D = 2, 3 and conjectured for D > 3 in [22,23]). The relations fulfilled by x i , L hk are determined in section 4: the commutators [x i , x j ] are also Snyder-like [1], i.e. are proportional to L hk /k, with a proportionality factor that is the same constant on all of H Λ , except on the l = Λ component of the latter, where it is a slightly different constant. x i generate the whole A Λ . The square distance x 2 ≡ x i x i is a function of L 2 only, such that almost all its spectrum is very close to 1 and goes to 1 in the limit Λ → ∞. In section 6 we show in which sense H Λ , A Λ go to H, A as Λ → ∞, in particular how one can recover f · ∈ C(S d ) ⊂ A, the multiplication operator of wavefunctions in L 2 (S d ) by a continuous function f , as the strong limit of a suitable sequence f Λ ∈ A Λ (again, this had been only conjectured in [23]). In section 7 we discuss our results and possible developments in comparison with the literature; in particular, we point out that our pair (H Λ , A Λ ) can be seen as a fuzzy quantization of a coadjoint orbit of O(D) that can be identified with the cotangent space T * S d , the classical phase space over the d-dimensional sphere. Finally, we have concentrated most proofs in the appendix 8.

General setting
We choose a set of real Cartesian coordinates x := (x 1 , ...x D ) of R D and abbreviate ∂ i ≡ ∂/∂x i . We normalize x i , ∂ i and H itself so as to be dimensionless. Then we can express where actually x i = x i and ∂ i = ∂ i because the coordinates are real and Cartesian. The self-adjoint operators which are equivariant under all orthogonal transformations Q (including parity Q = −I) All scalars S, in particular S = ∆, r 2 , V, H, are invariant. This implies [S, L ij ] = 0, where are the angular momentum components associated to x. These generate rotations of R D , i.e.
hold for the components v h of all vector operators, in particular v h = x h , ∂ h , and close the commutation relations of so(D), The D derivatives ∂ i make up a globally defined basis for the linear space of smooth vector fields on R D . As the L ij are vector fields tangent to all spheres r =const, the set B = {∂ r , L ij | i < j} (∂ r := ∂/∂r) is an alternative complete set that is singular for r = 0, but globally defined elsewhere; for D = 2 it is a basis, while for D > 2 it is redundant, because of the relations This redundancy (unavoidable if S d is not parallelizable) will be no problem for our purposes. We shall assume that V (r) has a very sharp minimum at r = 1 with very large k = V (1)/4 > 0, and fix V 0 := V (1) so that the ground state ψ 0 has zero energy, i.e. E 0 = 0 (see fig. 2). We choose an energy cutoff E fulfilling first of all the condition so that we can neglect terms of order higher than two in the Taylor expansion of V (r) around 1 and approximate the potential as a harmonic one in the classical region v E ⊂ R D  determined by the condition V (r) ≤ E. By (13), v E is approximately the spherical shell |r − 1| ≤ E−V 0 2k ; when both E − V 0 and k diverge, while their ratio goes to zero, then v E reduces to the unit sphere S d . We expect that in this limit the dimension of H E diverges, and we recover standard quantum mechanics on S d . As we shall see, this is the case.
Of course, the eigenfunctions of H can be more easily determined in terms of polar coordinates r, θ 1 , ..., θ d , recalling that the Laplacian in D dimensions decomposes as follows (see section 3.1) where L 2 := L ij L ij /2 is the square angular momentum (in normalized units), i.e. the quadratic Casimir of U so(D) and the Laplacian on the sphere S d ; L 2 can be expressed in terms of angles θ a and derivatives ∂/∂θ a only. The eigenvalues of L 2 are l (l + D − 2), see section 3.1; we denote by V l D the L 2 = E l eigenspace within L 2 (S d ).
.., θ d ), transforms the Schrödinger PDE Hψ = Eψ into the Fuchsian ODE in the unknown g(r) 1 The requirement ψ ∈ L 2 (R D ) implies that g belongs to L 2 (R + ), in particular goes to zero as r → ∞. The self-adjointness of H implies that it must be f (0) = 0; this is compatible [23] with Fuchs theorem provided r 2 V (r) (13)]. Since V (r) is very large outside the thin spherical shell v E (a neighbourhood of S d ), g, f, ψ become negligibly small there, and, by condition (13), the lowest eigenvalues E are at leading order those of the 1-dimensional harmonic oscillator approximation [23] of (15) which is obtained neglecting terms O (r−1) 3 in the Taylor expansions of 1/r 2 , V (r) about r = 1. Here The (Hermite functions) square-integrable solutions of (16) (here M n,l are normalization constants and H n are the Hermite polynomials) lead to The corresponding 'eigenvalues' in (16) E n,l = (2n + 1) √ k l lead to energies As said, we fix V 0 requiring that the lowest one E 0,0 be zero; this implies , and the expansions of E n,l and r l at leading order in k become E 0,l coincide at lowest order with the desired eigenvalues E l of the Laplacian L 2 on S d , while if n > 0 E n,l diverge as k → ∞; to exclude all states with n > 0 (i.e., to 'freeze' radial oscillations; then all corresponding classical trajectories are circles) we impose the cutoff The right inequality is satisfied prescribing a suitable dependence k (Λ), e.g. k (Λ) = [Λ(Λ+D−2)] 2 ; the left one is satisfied setting n = 0 and l ≤ Λ. Abbreviating f l ≡ f 0,l , we end up with eigenfunctions and associated energies (at leading order in 1/Λ) Thus H Λ decomposes into irreps of O(D) (and eigenspaces of L 2 , H) as follows We can express the projectors P l Λ : H Λ → H l Λ as the following polynomials in L 2 : In the commutative limit Λ → ∞ the spectrum {E 0,l } Λ l=0 of H goes to the whole spectrum {E l } l∈N 0 of L 2 . If φ, φ ∈ H ≡ L 2 (R D ) can be factorized into radial parts f (r), f (r) and angular parts T, T ∈ L 2 (S d ), i.e. φ = f T , φ = f T , then so can be their scalar product: Here we have denoted by ·, · the scalar product of L 2 (S d ), where dα is the O(D)-invariant measure on S d2 . Assume that B := {Y m l } (l,m)∈I , is an orthonormal basis of L 2 (S d ) consisting of eigenvectors of L 2 (e.g. spherical harmonics), here m ∈ I l is a (multi-)index 3 labelling the elements of an orthonormal basis B l ≡ {Y m l } m∈I l of V l D , and I := {(l, m) | l ∈ N 0 , m ∈ I l }. Then, by appropriate choices of the normalization constants 4 of (18), one obtains as orthonormal bases respectively of H and H l Λ B := ψ m n,l := f n,l (r)Y m l , | n ∈ N 0 , l ∈ I , The projector P l Λ : H → H l Λ acts by If φ has the form φ(r, θ) = Θ j (θ)φ(r), with Θ j ∈ V j D , then by (24), (26) this simplifies to 2 In terms of the angles θ, dα = sin d−1 (θ d ) sin d−2 (θ d−1 ) · · · sin (θ 2 ) dθ 1 dθ 2 · · · dθ d . , which has zero asymptotic expansion in 1/ √ 2k, see [22].
which is zero if l = j, has the same angular dependence Θ j (θ) as φ if l = j. In next section we provide an explicit characterization of elements Θ l ∈ V l D as polynomials in the coordinates t i of points of the unit sphere S d , which fulfill the relation rather than as combinations of spherical harmonics Y m l (θ), m ∈ I l .
3 Representations of O(D) via polynomials in x i , t i The differential operator L 2 can be expressed as The 'dilatation operator' η and the Laplacian ∆ fulfill In particular, the action of η on monomials in the x i amounts to multiplication by their total degree. In terms of polar coordinates it is η = r∂ r , which replaced in (29) gives (14).
Let C[x 1 , ..., x D ] be the space of complex polynomial functions on R D and, for all l ∈ N 0 , let W l D be the subspace of homogeneous ones of degree l. The monomials of degree l x i 1 x i 2 ...x i l ∈ W l D can be reordered in the form (x 1 ) l 1 ...(x D ) l D and make up a basis of W l D : the dimension of W l D is the number of elements of B W l D . Clearly W l D carries a representation of O(D) as well as U so(D), but this is reducible if l ≥ 2; in fact, the subspace r 2 W l−2 D ⊂ W l D manifestly carries a smaller representation. We denote byV l D the "trace-free" component of W l D , namely the subspace such that W l D =V l D ⊕ r 2 W l−2 D . As a consequence, V l D carries the irreducible representation (irrep) π l D of U so(D) and O(D) characterized by the highest eigenvalue of L 2 within W l D , namely E l (the eigenvalues of all other [D/2−1] Casimirs are determined by l). Abbreviating X hk l,± := (x h ±ix k ) l , this can be easily shown observing that for all h, k ∈ {1, ..., D} X hk l,± ∈ W l D are annihilated by ∆ and are eigenvectors of L 2 with that eigenvalue; moreover, they are eigenvectors of L hk with eigenvalue ±l. Hence X hk l,+ , X hk l,− can be used as the highest and lowest weight vectors ofV l D 5 . Since all the L ij commute with ∆,V l D can be characterized also as the subspace of W l D which is annihilated by ∆. A complete set inV l D consists of trace-free homogeneous polynomials X i 1 i 2 ...i l l , which we will obtain below applying the completely symmetric trace-free projector P l to the x i 1 x i 2 ...x i l 's.
We slightly enlarge C[x 1 , ...x D ] introducing as new generators r, r −1 subject to the relations r 2 = x i x i (sum over i), rr −1 = 1. Inside this enlarged algebra the elements fulfill the relation (28) characterizing the coordinates of points of the unit sphere S d . Choosing g(r) = r −l in (33-36) we obtain the same relations with x i , X hk l,± replaced by t i , T hk l,± := (t h ±it k ) l . We shall denote by P ol D the algebra of complex polynomials in such t i , by P ol Λ D the subspace of polynomials up to degree Λ, by P Λ : P ol D → P ol Λ D the corresponding projector. P ol D endowed with the scalar product T, T := S d dα T * T is a pre-Hilbert space; its completion is L 2 (S d ). We extend P Λ to all of L 2 (S d ) by continuity in the norm of the latter. Also P ol Λ D , V l D are Hilbert subspaces of In fact, in terms of Cartesian coordinates, using (29), (31) we immediately find the following commutation relations among operators of multiplication (x h ±ix k )· and differential operators ∆ X hk l,± = X hk l,± ∆ + 2lX hk l−1,± (∂ h +i∂ k ), Consequently, we obtain the following functions at the rhs as results of the operator actions on functions at the lhs: ∆X hk l,± = 0 and, for all functions g(r), L hk X hk l,± g(r) = ±l X hk l,± g(r), L hk X hk l,+ X hk m,− g(r) = (l−m) X hk l,+ X hk m,− g(r).
Denoting by τ = D 2 the rank of so(D), as a basis of a Cartan subalgebra of so(D) one can take any set into irreps carried by V l D :=V l D /r l . Its dimension is thus We have proved the first isomorphism in section 2 and will prove the second in section 3.2.

O(D)-irreps via trace-free completely symmetric projectors
can be seen as the set of components of an element of E with respect to (w.r.t.) an orthonormal basis. The permutator on E ⊗ 2 ≡ E ⊗E is defined via P(u⊗v) = v⊗u and linearly extended. In all bases it is represented by the D 2 × D 2 matrix P hi jk = δ h k δ i j . The symmetric and antisymmetric projectors P + , P − on E ⊗ 2 are obtained as here and below we denote by 1 D l the identity operator on E ⊗ l , which in all bases is represented , while the symmetrized one P + E ⊗ 2 contains two irreps: the 1-dim trace one and the trace-free symmetric one. The matrix representation of the 1-dim projector P t on the former is where the D × D metric matrix g ij (in the chosen basis) is a so(D)-isotropic symmetric tensor, and g ij g jh = δ i h , whence g ij g ij = D. Here we shall use an orthonormal basis of E, whence g ij = g ij = δ ij , and indices of vector components can be raised or lowered freely, e.g.
-dim trace-free symmetric projector P s is given by These projectors satisfy the equations where α, β = −, s, t. In the sequel we shall abbreviate P ≡ P s . This implies in particular PP = 0, where we have introduced the new projector P := P − + P t . P, P t are symmetric matrices, i.e. invariant under transposition T , and therefore also the other projectors are: Given a (linear) operator M on E ⊗ n , for all integers l, h with l > n, and 1 ≤ h ≤ l +1−n we denote by M h(h+1)...(h+n−1) the operator on E ⊗ l acting as the identity on the first h−1 and the last l+1−n−h tensor factors, and as M in the remaining central ones. For instance, if M = P and l = 3 we have P 12 = P ⊗ 1 D , P 23 = 1 D ⊗ P. It is straightforward to check Proposition 3.1 All the projectors A = P + , P − , P, P t , P fulfill the "braid" relation Moreover, Proof Since A = 1 D 2 , P, P t fulfill (45), then also A = P + , P − , P, P t , P do. One can immediately check the first equality in (148) via direct calculation; left multiplying the first by P 12 one obtains the second. Eq. (47) are obtained from (148) exchanging 1 ↔ 3 and using the symmetry of P, P t under the flip. Eq. (48) are obtained from (47) by transposition.
Next, we define and determine the completely symmetric trace-free projector P l on E ⊗ l generalizing P 2 ≡ P to l > 2. It projects the tensor product of l copies of E to the carrier space of the l-fold completely symmetric irrep of U so(D), isomorphic to V l D , therein contained. It is uniquely characterized by the following properties: Consequently, it is also tr 1...l P l = dim(V l D ), which guarantees that P l acts as the identity (and not as a proper projector) on V l D . The right relations in (50) amount to P li1...il j 1 ...j l δ jnj n+1 = 0, δ ini n+1 P li1...il j 1 ...j l = 0, n = 1, ..., l−1.
Clearly the whole of (50) can be summarized as P l P n(n+1) = 0 = P n(n+1) P l . It is straightforward to prove that the above properties imply also the ones Proposition 3.2 The projector P l+1 can be expressed as a polynomial in the permutators P 12 , ..., P (l−1)l and trace projectors P t 12 , ..., P t (l−1)l through either recursive relation As a consequence, the P l are symmetric, (P l ) T = P l .
This the analog of Proposition 1 in [26] for the quantum group U q so(D) covariant symmetric projectors; the proof is in the appendix. By a straightforward computation one checks that (h is summed over). Using (31), (52) one easily shows that the homogeneous polynomials are harmonic, i.e. satisfy ∆X i 1 ...i l l = 0; using (29), we find that they are eigenvectors of L 2 , with eigenvalues (3). They make up a complete set inV l D , which can be thus also characterized as the subspace of W l D that is annihilated by ∆, whereas ∆φ = 0 for all φ ∈ r 2 W l−2 D . The X i 1 ...i l l are not all independent, because they are invariant under permutations of (i 1 ...i l ) and by (52) fulfill the linear dependence relations Proposition 3.3 In a compact notation, The proof is in Appendix 8.2. More explicitly, (61) becomes Contracting the previous relation with δ hi 1 and using (52) we obtain In the Appendix we also prove Proposition 3.4 The maps L hk :V l D →V l D explicitly act as follows: Dividing (59), (60), (62), (63), (64) by the appropriate powers of r we find (65) they make up a complete set T l in it, but not a basis, because they are invariant under permutations of (i 1 ...i l ) and by (60) fulfill the linear dependence relations The actions of the operators t h ·, iL hk on the T i 1 ...i l l explicitly read be its decompositions in the basis of spherical harmonics and in the complete set T := ∞ l=0 T l ; here the two sets of coefficients are related by φ l ..i l are uniquely determined if, as we shall assume, we choose them trace-free and completely symmetric, i.e. fulfilling whence φ l i 1 ...i l δ ini n+1 = 0 for n = 1, ..., l−1. Then (70) can be also written in the form The projector P Λ acts by truncation, All completely symmetric, O(D)-isotropic tensors of even rank N are proportional to The .
In terms of the decompostions (70b)-(71) the scalar product of generic φ, ψ ⊂ H s is equal to In particular, (76) implies ..i l and more generally the second equality holds only if (71) holds. Then, we have also We now determine the decomposition of decomposes as follows into V n D components: where L lm := {|l−m|, |l−m|+2, ..., l+m} and, defining r := l+m−n 2 ∈ {0, 1, ..., m}, The coefficients S i 1 ...i l ,j 1 ...jm are the analogs of the Clebsch-Gordon coefficients, which appear in the decomposition of a product of two spherical harmonics into a combination of spherical harmonics for D = 3. The first term of the sum (79) is T i 1 ...i l j 1 ...jm l+m . This is consistent with the first term in the iterated application of (67). If r = m = 1, since P 1j 1 a 1 = δ j 1 a 1 , n = l−1, the result is consistent with the second term in (67):

Embedding in
as the subgroup of SO(D) which is the little group of the D-th axis; its Lie algebra, isomorphic to so(D), is generated by the L hk . We shall add D as a subscript to distinguish objects in this enlarged dimension from their counterparts in dimension D, e.g. the distance r D from the origin in R D , from its counterpart r ≡ r D in R D , P l D from P l ≡ P l D , and so on. We look for the decomposition of eachV Λ D into irreps of such a U so(D). Clearly,V 0 where the projectors P l D are constructed as in Proposition 3.2, but with D replaced by D. Any pair of indices I a , I b appears either through the product x Ia x I b or through r 2 D δ IaI b . If we introduce r D as a further generator constrained by the relation r 2 D = x I x I (sum over I), then the X I 1 ...I Λ D,Λ can be seen also as homogeneous polynomials of degree Λ in x I , r D . Since δ jaD = 0, in X j 1 ...j l D...D D,Λ : any pair of indices DD appears either through the product x D x D or through r 2 D δ DD = r 2 D ; any pair of indices j a , D appears through the product x ja x I b ; any pair of indices j a , j b appears either through the product x ja x j b or through r 2 D δ jaj b . By property (52), the latter terms completely disappear in any combination P l i 1 ...i l j 1 ...j l X j 1 ...j l D...D D,Λ , l ∈ {2, 3, ..., Λ}. Therefore such a combination can be factorized as followš wherep Λ,l is a homogeneous polynomial of degree Λ − l in x D , r D of the form Proposition 3.8V Λ D decomposes into the following irreducible components of U so(D): whereV l D,Λ V l D is spanned by theF i 1 ...i l D,Λ , since the latter are eigenvectors of L 2 : Denoting by [a] the integral part of a ∈ R + , the coefficients of (84) are given by , and under L hD as follows: The proof is in Appendix 8.5. We now determine the decomposition of V Λ The F ij D,2 can be expressed as combinations of the T IJ D,2 : where we have introduced a polynomial p Λ,l of degree h = Λ−l in t D (containing only powers of the same parity as h) by p Λ,l :=p Λ,l (x D , r D ) r l−Λ D ; more explicitly the latter reads As a direct consequence of Propositions 3.8, dividing all relations by (r D ) Λ , we find Corollary 3.9 V Λ D decomposes into the following irreducible components of U so(D): where V l D,Λ V l D is spanned by the F i 1 ...i l D,Λ . The latter are eigenvectors of L 2 , transform under L hk as the T i 1 ...i l l , and under L hD as follows: For convenience, we slightly enlarge U so(D) by introducing the new generator On the T i 1 ...i l l (spanning P ol Λ D ) and on generators L hi , P Λ t i · of End P ol Λ D they act bŷ wherê .
The proposition and its proof are obtained from Theorem 5.1 and the associated proof by fixing Λ, taking k independent of Λ and letting the k → ∞.
By (27), (67), applying up to order O k −3/2 , see appendix 8.6. Hence, at the same order, The O k −3/2 corrections depend on the terms proportional to (r−1) k , k > 2, in the Taylor expansion of V . These could be set rigorously equal to zero by a suitable choice of V . Henceforth we adopt (101-103) as exact definitions of L hk , x i . In the appendix we prove This is the analog of Proposition 4.1 in [22]. We obtain a fuzzy sphere choosing k as a function k(Λ) fulfilling (20), e.g. k = Λ 2 (Λ+D−2) 2 /4; the commutative limit is Λ → ∞.

Remarks:
4.a Eq. (106) is the analog of (12). By (108), it can be reformulated also in the form is not a constant, but by (109), (23) l=Λ can be expressed as a polynomial χ in L 2 only, with the same eigenspaces H l Λ . All its eigenvalues r 2 l , except r 2 Λ , are close to 1, slightly (but strictly) grow with l and collapse to 1 as Λ → ∞. Conversely, L 2 can be expressed as a polynomial υ in x 2 , via L 2 = Λ l=0 E l P l Λ and P l Λ = Λ n=0,n =l 4.c By (108), (23) l=Λ the commutators [x i , x j ] are Snyder-like, i.e. of the form αL ij ; also α depends only on the L hk , more precisely can be expressed as a polynomial in L 2 .
4.d Using (104), (105), (108), all polynomials in x i , L hk can be expressed as combinations of monomials in x i , L hk in any prescribed order, e.g. in the natural one the coefficients, which can be put at the right of these monomials, are complex combinations of 1 and P Λ Λ . Also P Λ Λ can be expressed as a polynomial in L 2 via (23) l=Λ .
Hence a suitable subset (depending on Λ) of such ordered monomials makes up a basis of the N 2 -dim vector space A Λ .
4.e Actually, x i generate the * -algebra A Λ , because also the L ij can be expressed as nonordered polynomials in the x i : by (108) L ij = [x j , x i ]/α, and also 1/α, which depends only on P Λ Λ , can be expressed itself as a polynomial in x 2 , as shown above.
5 Isomorphisms of H Λ , A Λ , and * -automorphisms of A Λ On the ψ i 1 ...i l l (spanning H Λ ) and on generators L hi , x i · of A Λ they respectively act as follows: where X i := L Di , p Λ,l = p Λ,l t D are the polynomials (90), and here A := k + (D−1)(D−3)3/4, and Γ is Euler gamma function.
The proof is in appendix 8.8. The theorem extends Propositions 3.2, 4.2 of [22] to d > 2.
The claims for d > 2 were partially formulated, but not proved, in [23].
As already recalled, the group of * -automorphisms of A Λ M N (C) is inner and isomorphic to SU (N ), i.e. of the type 6 Fuzzy spherical harmonics, and limit Λ → ∞ In this section we suppress Einstein's summation convention over repeated indices. The previous results allow to define U so(D)-module Hilbert space isomorphisms with suitable coefficients N lm n related to their classical limits N lm n > 0 of formula (79) by As a fuzzy analog of the vector space C(S d ) we adopt here the highest l is 2Λ because by (123) the T i 1 ...i l l annihilate H Λ if l > 2Λ. By construction, is the decomposition of C Λ into irreducible components under O(D). V l D is trace-free for all l > 0. In the limit Λ → ∞ (125) becomes the decomposition of C(S d ). As a fuzzy analog of f ∈ C(S d ) we adopt the sumf 2Λ appearing in (124) with the coefficients of the expansion (70) of f up to l = 2Λ. In appendix 6.2 we prove Theorem 6.2 For all f, g ∈ C(S d ) the following strong limits as Λ → ∞ hold: The last statement says that the product in A Λ of the approximations f 2Λ , g 2Λ goes to the product in B (H s ) (the algebra of bounded operators on H s ≡ L 2 (S d )) of f ·, g·. We point out thatf 2Λ does not converge to f in operator norm, because the operatorf 2Λ (a polynomial in the x i ) annihilates H ⊥ Λ (the orthogonal complement of H Λ ), since so do the x i = P Λ x i · P Λ . Essentially the same claims of this theorem were proved for d = 1, 2 in [22] requiring that k(Λ) diverges much faster than required by (20), and were formulated (without proof) for d > 2 in Theorem 7.1 of [23] with the same strong assumptions on the divergence of k(Λ).

Outlook, discussion and conclusions
In this paper we have completed our construction [22,23] of fuzzy spheres S d Λ that be equivariant under the full orthogonal group O(D), D ≡ d+1, for all d ∈ N. The construction procedure consists (sections 1, 2) in starting with a quantum particle in R D configuration space subject to a O(D)-invariant potential V (r) with a very sharp minimum on the sphere of radius r = 1 and projecting the Hilbert space H = L 2 (R D ) to the subspace H E with energy below a suitable cutoff E; E is sufficiently low to exclude all excited radial modes of H (this can be considered as a quantum version of the constraint r = 1), so that on H E the Hamiltonian essentially reduces to the square angular momentum L 2 (the Laplacian, i.e. the free Hamiltonian, over the sphere S d ). By making both the confining parameter k ≡ V (1)/4 and E depend on Λ ∈ N, and diverge with it, we have obtained a sequence {(H Λ , A Λ )} Λ∈N of O(D)-equivariant approximations of a quantum particle on S d . H Λ is the Λ-th projected Hilbert space of states and A Λ ≡End(H Λ ) is the associated * -algebra of observables. The projected Cartesian coordinates x i no longer commute (section 4); their commutators [x i , x j ] are of Snyder type, i.e. proportional to the angular momentum components L ij . A Λ is spanned by ordered monomials (110) in x i , L ij (of appropriately bounded degrees), in the same way as the algebra A s of observables on H s is spanned by ordered monomials in t i , L ij . However, while x i generate the whole A Λ because [x i , x j ] ∝ L ij , this has no analog A s . The square distance x 2 from the origin is not identically 1, but a function of L 2 such that its spectrum is very close to 1 and collapses to 1 as Λ → ∞. We have also constructed (section 6) the subspace C Λ ⊂ A Λ of completely symmetrized trace-free polynomials in the x i ; this is also spanned by the fuzzy analogs of spherical harmonics (thought as multiplication operators on A basis of A Λ consists of a suitable (Λ-dependent) subset S Λ of ordered monomials (110). Since L ij ψ 0 = 0 for all i, j ≤ D, the subset S Λ of S Λ with all n ij = 0 is a basis of C Λ , and H Λ = C Λ ψ 0 ; H Λ , C Λ , P ol Λ D carry the same reducible representation of O(D). As Λ → ∞: i) S Λ becomes a basis S of A s consisting of ordered monomials in t h , L ij ; ii) S Λ becomes a basis S of C(S d ) consisting of ordered monomials in t h ; iii) S Λ ψ 0 becomes a basis of The structure of the pairs (H Λ , A Λ ) is made transparent by the discovery (section 5) that these are isomorphic to V Λ D , π Λ [U so(D)] , D ≡ D+1, also as O(D)-modules; π Λ is the irrep of U so(D) on the space V Λ D of harmonic polynomials of degree Λ on R D , restricted to S D . If we reintroduce and the physical angular momentum components l ij := L ij , and we define as usual the quantum Poisson bracket as {f, g} = [f, g]/i , then in the → 0 limit A s goes to the (commutative) algebra F of (polynomial) functions on the classical phase space T * S d , which is generated by t i , l ij . We can directly obtain F from A Λ adopting a suitable Λ-dependent going to zero as Λ → ∞ 6 . Using the isomorphism A Λ π Λ D U so(D) , we now show that, more formally, we can see {A Λ } Λ∈N as a fuzzy quantization of a coadjoint orbit of O(D) that goes to the classical phase space T * S d . We recall that given a Lie group G, a coadjoint orbit O λ , for λ in the dual space g * of the Lie algebra g of G, may be defined either extrinsically, as the actual orbit Ad * G λ of the coadjoint action Ad * G inside g * passing through λ, or intrinsically as the homogeneous space G/G λ , where G λ is the stabilizer of λ with respect to the coadjoint action (this distinction is worth making since the embedding of the orbit may be complicated). Coadjoint orbits are naturally endowed with a symplectic structure arising from the group action. If G is compact semisimple, identifying g * with g via the (nondegenerate) Killing form, we can resp. rewrite these definitions in the form which is also the dimension of T * S d , the cotangent space of the d-dimensional sphere S d (or phase space over S d ). This is consistent with the interpretation of A Λ as the algebra of observables (quantized phase space) on the fuzzy sphere. It would have not been the case if we had chosen some other generic irrep of U so(D): the coadjoint orbit would have been some other equivariant bundle over S d [27]. For instance, the 4-dimensional fuzzy spheres introduced in [16], as well as the ones of dimension d ≥ 3 considered in [17,18,19], are based on End(V Λ ), where the V Λ carry irreducible representations of both Spin(D) and Spin(D + 1), and therefore of both U so(D) and U so(D). Then: i) i for some Λ these may be only projective representations of O(D); ii) in general (106) will not be satisfied; iii) as Λ → ∞ V Λ does not go to L 2 (S d ) as a representation of U so(D), in contrast with our H Λ V Λ D . The X i := L iD play the role of fuzzy coordinates. As x 2 ≡ X i X i is central, it can be set x 2 = 1 identically. The commutation relations are also O(D)-covariant and Snyder-like, except for the case of the Madore-Hoppe fuzzy sphere [12,13]. The corresponding coadjoint orbit for d = 4 is the 6-dimensional CP 3 [20,21], which can be seen as a so(5)-equivariant S 2 bundle over S 4 (while [17] does not identify coadjoint orbits for generic d).
In [20,21] the authors consider also constructing a fuzzy 4-sphere S 4 N through a reducible representation of U so(5) on a Hilbert space V obtained decomposing an irrep π of U so(6) characterized by a highest weight triple (N, 0, n) with respect to (H λ 1 , H λ 1 H λ 1 ), where The X i = L i6 (i = 1, ..., 5), which make up a basis of the vector space so(6)\so (5), still play the role of noncommuting Cartesian coordinates. If n = 0 then x 2 ≡ 1 (V carries an irrep of O(5)), and one recovers the (so(5)-equivariant S 2 bundle over the) fuzzy 4-sphere of [16]. If n > 0, then the O(5)-scalar x 2 = X i X i is no longer central, but its spectrum is still very close to 1 provided N n, because then V decomposes only in few irreducible SO(5)-components, all with eigenvalues of x 2 very close to 1. The associated coadjoint orbit is 10-dimensional and can be seen as a so(5)-equivariant CP 2 bundle over CP 3 , or a so(5)-equivariant twisted bundle over either S 4 N or S 4 n . On the contrary, with respect to (H λ 1 , H λ 1 H λ 1 ) the highest weight triple of the irrep V Λ 6 considered here is (Λ, Λ, Λ); as said, x 2 ≡ x i x i 1 is guaranteed by adopting as noncommutative Cartesian coordinates the x i = m Λ (L 2 ) X i m Λ (L 2 ), with a suitable function m Λ , rather than the X i , and the associated coadjoint orbit has dimension 8, which is also the dimension of T * S 4 , as wished.
We now clarify in which sense we have provided a O(D)-equivariant fuzzy quantization of T * S d and S d -the phase space and the configuration space of our particle.
Although A s is generated by all the t h , L ij with h ≤ D, i < j ≤ D (subject to the relations (10), (11), (28), ε i 1 i 2 i 3 ....i D t i 1 L i 2 i 3 = 0 due to (12), t i t h = t h t i ), and C(S d ) is generated by the t h alone, the x i (or the simpler generators X i ) alone generate 7 the whole A Λ π Λ D U so(D) , which contains C Λ as a proper subspace, but not as a subalgebra. Thus the Hilbert-Poincaré series of the algebra generated by the x i (or X i ), A Λ , is larger than that of P ol Λ D and C Λ . If by a "quantized space" we understand a noncommutative deformation of the algebra of functions on that space preserving the Hilbert-Poincaré series, then {A Λ } Λ∈N is a (O(D)-equivariant, fuzzy) quantization of T * S d , the phase space on S d , while {C Λ } Λ∈N is not a quantization of S d , nor are the other fuzzy spheres, except the Madore-Hoppe fuzzy 2-dimensional sphere: all the others, as ours, have the same Hilbert-Poincaré series of a suitable equivariant bundle on S d , i.e. a manifold with a dimension n > d (in our case, n = 2d). (Incidentally, in our opinion also for the Madore-Hoppe fuzzy sphere the most natural interpretation is of a quantized phase space, because the → 0 limit of the quantum Poisson bracket endows its algebra with a nontrivial Poisson structure.) Therefore we understand H Λ , C Λ as fuzzy "quantized" S d in the following weaker sense. H Λ is the quantization of the space L 2 (S d ) of square integrable functions, and the space C Λ of fuzzy spherical harmonics is the quantization of the space C(S d ) of continuous functions, seen as operators acting on the former, because the whole H Λ is obtained applying to the ground state ψ 0 (or any other ψ ∈ H Λ ) the polynomials in the x i alone, or equivalently (by Proposition 5.1) the polynomials in the X i = L Di alone, or the space C Λ , in the same way as the Hilbert L 2 (S d ) is obtained (modulo completion) by applying C(S d ) or P ol D , i.e. the polynomials in the t i = x i /r, to the ground state, i.e. the constant function on S d . Many aspects of these new fuzzy spheres deserve investigations: e.g. space uncertainties, optimally localized states and coherent states also for d > 2, as done in [28,29,24] for d = 1, 2; a distance between optimally localized states (as done e.g. in [30] for the FS); extending the construction to particles with spin 8 ; QFT based on our fuzzy spheres; application of our fuzzy spheres to problems in quantum gravity, or condensed matter physics; etc. It would be also interesting to investigate whether our procedure can be applied (or generalized) to other symmetric compact submanifolds 9 S ⊂ R D that are level sets of smooth or polynomial function(s) ρ(x).
Finally, we point out that a different approach to the construction of noncommutative submanifolds of noncommutative R D , equivariant with respect to a 'quantum group' (twisted Hopf algebra) has been proposed in [32,31]; it is based on a systematic use of Drinfel'd twists.
We first determine the coefficients β l+1 , γ l+1 by imposing the conditions (50). By the recursive assumption, only the condition with m = l is not fulfilled automatically and must be imposed by hand. Actually, it suffices to impose just (50a), due to the symmetry of the Ansatz (54) and of the matrices P l under transposition. Abbreviating M ≡ M (l) this amounts to where we have used also the relations = DP t l(l+1) P t (l−1)l P l(l+1) P l(l+1) The conditions that the three square brackets vanish are recursively solved, starting from l = 1 with initial input β 1 = 0 = γ 1 (since P 1 = 1 D ), by We determine the coefficient α l+1 by imposing the condition (51). This gives The condition that the square bracket vanishes is recursively solved, starting from l = 0 with initial input α 0 = 1, by α l+1 = 1/(l +1). This makes (128) into (56) [yielding back (42) if l = 2]. We have thus proved that the Ansatz (54) fulfills (50-51). Similarly one proves that also the Ansatz (55) does the same job.

Proof of Proposition 3.4
Using Proposition 3.3 we easily find Alternatively, from (133) we obtain also (64), because Multiplying the previous relations by 1/r l , which commutes with L hk , (64) give (69).

Proof of Proposition 3.6 and Theorem 3.7
The right-hand side (rhs) of (73) is the sum of N ! terms; in particular, G ij 2 = δ ij + δ ji = 2δ ij . G N , G N −2 are related by the recursive relation The rhs is the sum of N (N−1) products δ ··· G ··· N −2 . The 'trace' of G N equals tr (G 2 ) = 2D for N = 2 and by (134) fulfills the recursive relation tr(G N ) = N (D+N−2) tr(G N −2 ). In fact, each (134)  ; the sum has (2l)! terms. In fact, all terms where both indices of at least one Kronecker δ contained in G 2l are contracted with the two indices of the coefficients φ l i 1 ...i l , or of the ψ l i 1 ...i l , vanish, by (71). The remaining 2 l (l!) 2 terms arise from the (l!) 2 products contained in G 2l of the type δ i π(1) j π (1) ...δ i π(l) j π (l) , where π, π are permutations of (1, ..., l), and the other ones which are obtained exchanging the order of the indices i π(h) j π (h) → j π (h) i π(h) in one or more of these Kronecker δ's; they are all equal, again by (71). Hence, By the orthogonality V l D ⊥ V l D for l = l we find that the scalar product of generic φ, ψ ⊂ H s is given by (76). This concludes the proof of Proposition 3.6.
Applying m times (67) and absorbing into a suitable combination of T n 's (n ∈ L) the m-degree monomials in t i whose combination gives T j 1 ...jm m , we find that (79) must hold with suitable coefficients S i 1 ...i l ,j 1 ...jm k 1 ...kn ∈ R. We determine these coefficents faster using (79) as an Ansatz, making its scalar product with T k 1 ...
where N = l+m+n = n+r =: 2s is even. Due to the form of G N , the sum has N ! terms, each containing a product of s Kronecker δ's. All terms where both indices of some Kronecker δ contained in G 2s are contracted with two indices of P l , or P m , or P n , vanish, by (52). As a warm-up, consider first the case n = l+m. Renaming for convenience b 1 , ..., b m as a l+1 , ...a n , the remaining 2 n (n!) 2 terms arise from the (n!) 2 products of the type δ c π(1) a π (1) ... δ c π(n) a π (n) contained in G N , where π, π are permutations of (1, ..., h), and the other ones which are obtained exchanging the order of the indices c π(h) a π (h) → a π (h) c π(h) , in one or more of these n Kronecker δ's; by the complete symmetry of P n , they are all equal to the term where π, π are the trivial permutations; correspondingly, the product is δ c 1 a 1 ... δ cnan . Hence, . This is consistent with the first term in the (even iterated) application of (67). For generic n ∈ L (136) becomes Since L hk commute with scalars, and [L hk , x D ] = 0,p Λ,l commutes with all the L hk and therefore also with L 2 . Hence, using (59), we find L 2p Λ,l X i 1 ...i l l =p Λ,l L 2 X i 1 ...i l l = E lpΛ,l X i 1 ...i l l , i.e (86). To compute the coefficients b Λ,l+2k we preliminarly note that  namely, more compactly, we obtain (87). TheF i 1 ...i l D,Λ transform as the X i 1 ...i l l under the action of the L hk , because the latter commute withp Λ,l . Using (7), (83), and the fact that ∂ D annihilates all polynomials in the x i , we find (equalities (138-139) are proved by direct calculations), whence (88).

8.6
Evaluating a class of radial integrals, and proof of (103) Given a smooth h(r) not depending on k, formula (98) of [22] gives the (asymptotic expansion of) the radial integral of its product with g l (r)g L (r) (l, L ∈ N 0 ) at lowest order in 1/k: r l , k l were defined in (17), while r l,L := Up to O k −3/2 the exponential in (140) is 1, because by explicit computation By (27), (67), applying The last integral is of the type (140) with L = l−1 and h(r) = r. By explicit computation, By setting the O k −3/2 term equal to zero we finally arrive at (102-103).

Proof of Proposition 5.1
We first show that indeed A Λ is generated by the L hi , x i ·. Since the action of so(D) (which is spanned by the L hi ) is transitive on each irreducible component H l Λ V l D (l ∈ {0, 1, ..., Λ}) contained in H Λ , it remains to show that some ψ l ∈ H l Λ can be mapped into some element of H m Λ for all m = l by applying polynomials in by the contracted multiplication of (68), while applying (x 1 + ix 2 )· to (t 1 + it 2 ) l f l (r) ∈ H l Λ one obtains a vector proportional to (t 1 +it 2 ) l+1 f l+1 (r) ∈ H l+1 Λ . The Ansatz (113) with generic coefficients a Λ,l is manifestly U so(D)-equivariant, i.e. fulfills (112) for all a = L hi ; it is also invariant under permutations of (i 1 ...i l ) and fulfills relations (100) (both sides give zero when contracted with any δ iai b ). More explicitly, (113) read where we have abbreviated µ Λ (l) ≡ m Λ (l) m * Λ (l +1). We determine the unknown m Λ , a Λ,l requiring (112) for a = x h ·. Applying κ Λ to eq. (103) with l < Λ, and imposing (112) we obtain c l+1 a Λ,l+1 F hi 1  , which implies the recursion relations for the coefficients a Λ,l c l+1 a Λ,l+1 = i(Λ−l)a Λ,l µ Λ (l), c l a Λ,l−1 = −i(Λ+l+D−2)a Λ,l µ * Λ (l−1).

Proof of Theorem 6.1
The proof is recursive. By (119) T The last equality holds because the sum is nothing but f 2Λ φ Λ 2 , i.e. what we would obtain from (150) replacing N lm n → 0, and thereforef 2Λ → 0. Moreover, f 2Λ φ Λ ≤ f 2Λ op φ Λ ; but both factors have Λ-independent bounds: φ Λ ≤ φ , while by the triangular inequality, f 2Λ op ≤ f op + f 2Λ − f op , and the second term goes to zero as Λ → ∞, hence is bounded by some η ≥ 0. So we end up with the limit is zero because by (20) k(Λ) diverges at least as Λ 4 , then by (111) Λ (Λ) goes to zero as Λ −1 . Replaced in (149) this yields for all f ∈ C(S d ) i.e. f 2Λ → f · strongly, as claimed. Replacing f → f g, we find also that (f g) 2Λ → (f g)· strongly for all f, g ∈ C(S d ), as claimed. On the other hand, since φ ⊥ Λ ≤ φ (because φ = φ Λ + φ ⊥ Λ with φ Λ , φ ⊥ Λ = 0), relations (149) and (149) imply also where F > 0 is an upper bound for the expression in the square bracket; hence i.e. the operator norms f 2Λ op of thef 2Λ are uniformly bounded: f 2Λ op ≤ (F + f ∞ ) φ . Therefore (153) implies the last claim of the theorem