The C -Algebras of the Heisenberg Group Hn, N ≥ 1, and the Thread-Like ∗ Lie Groups GN , N ≥ 3, in Terms of C -Algebras of Operator ﬁelds

The C∗-algebras of the Heisenberg Group and of thread-like Lie groups.

Jean Ludwig and Lyudmila Turowska

Abstract ∗ We describe the C -algebras of the Heisenberg group Hn, n ≥ 1, and the thread-like ∗ Lie groups GN , N ≥ 3, in terms of C -algebras of operator ﬁelds.

1 Introduction and notation

Let Hn be the Heisenberg group of dimension 2n + 1. It has been known for a long time that ∗ ∗ ∗ the C -algebra, C (Hn), of Hn is an extension of an ideal J isomorphic to C0(R , K) with ∗ 2n ∗ the quotient algebra isomorphic to C (R ), where K is the C -algebra of compact operators ∗ ∗ ∗ on a separable Hilbert space, R = R \{0} and C0(R , K) is the C -algebra of continuous ∗ functions vanishing at inﬁnity from R to K. ∗ 2n We obtain an exact characterization of this extension giving a linear mapping from C (R ) ∗ ∗ ∗ to C (Hn)/J which is a cross section of the quotient mapping i : C (Hn) → C (Hn)/J. More ∗ ∗ ∗ precisely, realizing C (Hn) as a C -subalgebra of the C -algebra Fn of all operator ﬁelds ∗ ∗ 2n (F = F (λ))λ∈R taking values in K for λ ∈ R and in C (R ) for λ = 0, norm continuous on ∗ ∗ 2n R and vanishing as λ → ∞, we construct a linear map ν from C (R ) to Fn, such that the ∗ ∗ C -subalgebra is isomorphic to the C -algebra Dν(Hn) of all (F = F (λ))λ∈R ∈ Fn such that

kF (λ) − ν(F (0))kop → 0, where k · kop is the operator norm on K. The constructed mapping ν is an almost homomorphism in the sense that

lim kν(f · h)(λ) − ν(f)(λ) ◦ ν(h)(λ)kop = 0. λ→0

∗ 2 ∗ Moreover, any such almost homomorphism τ : C (R ) → Fn deﬁnes a C -algebra, Dτ (Hn), ∗ ∗ 2n which is an extension of C0(R , K) by C (R ). A question we left unanswered : what mappings ∗ ∗ τ give the C -algebras which are isomorphic to C (Hn). We note that the condition

∗ 2n lim kτ(h)(λ)kop = khkC∗( 2n), for all h ∈ C (R ), λ→0 R

∗ which is equivalent to the condition that the topologies of Dτ (Hn) and that of C (Hn) agree, is not the right condition: there are examples of splitting extensions of type Dτ (Hn) with the ∗ ∗ same spectrum as C (Hn) (see [De] and Example 2.23) while it is known that C (Hn) is a non-splitting extension. ∗ ∗ We note that another characterisation of C (Hn) as a C -algebra of operator ﬁelds is given without proof in a short paper by Gorbachev [Gor].

1 ∗ The second part of the paper deals with the C -algebra of thread-like Lie groups GN , N ≥ 3. The group G3 is the Heisenberg group of dimension 3 treated in the first part of the paper. The groups GN are nilpotent Lie groups and their unitary representations can be described using the Kirillov orbit method. The topology of the dual space GdN has been investigated in details in [ALS]. In particular, it was shown that like for the Heisenberg group G3 the topology ∗ 2 of GdN , N ≥ 3 is not Hausdorff. It is known that Gc3 = R ∪ R as a set with natural topology ∗ 2 on each pieces, the limit set when λ ∈ R goes to 0 is the whole real plane R . The topology of GdN , becomes more complicated with growth of the dimension N. Using a description of the limit sets of converging sequences (πk) ∈ GdN obtained in [AKLSS] and [ALS] we give ∗ a characterisation of the C -algebra of GN in the spirit of one for the Heisenberg group gen 2 gen ∗ Hn. Namely, parametrising GdN by a set SN ∪ R , where SN consists of element ` ∈ gN ∗ corresponding to non-characters (here gN is the Lie algebra of GN ), we realize C (GN ) as ∗ gen gen a C -algebra of operator fields (A = A(`)) on SN ∪ {0}, such that A(`) ∈ K, ` ∈ SN , ∗ 2 A(0) ∈ C (R ) and (A = A(`)) satisfy for each converging sequence in the dual space the generic, the character and the infinity conditions (see Definition 3.12). p n We shall use the following notation. L (R ) denote the space of (almost everywhere equivalence classes) p-integrable functions for p = 1, 2 with norm k · kp. By kfk∞ we denote the supremum norm supx∈Ω |f(x)| of a continuous function f vanishing at infinity from a locally n ∞ compact space Ω to C. D(R ) is the space of complex-valued C functions with compact n support and S(R ) is the space of Schwartz functions, i.e. rapidly decreasing complex-valued ∞ n C functions on R . The space of Schwartz functions on the groups Hn and GN (see [CG]) will be denoted by S(Hn) and S(GN ) respectively. We use the usual notation B(H) for the space of all linear bounded operators on a Hilbert space H with the operator norm k · kop.

Keywords. Heisenberg group, thread-like Lie group, unitary representation, C∗-algebra.

2000 Mathematics Subject Classiﬁcation: 22D25, 22E27, 46L05.

∗ 2 The C -algebra of the Heisenberg group Hn

Let Hn be the 2n + 1 dimensional Heisenberg group, which is deﬁned as to be the Lie group n n whose underlying variety is the vector space R × R × R and on which the multiplication is given by 1 (x, y, t)(x0, y0, t0) = (x + x0, y + y0, t + t0 + (x · y0 − x0 · y)), 2 n where x · y = x1y1 + ··· + xnyn denotes the Euclidean scalar product on R . The center of Hn is the subgroup Z := {0n} × {0n} × R and the commutator subgroup [Hn,Hn] of Hn is given by [Hn,Hn] = Z. The Lie algebra g of Hn has the basis

B := {Xj,Yj, j = 1 ··· , n, Z = (0n, 0n, 1)}, where Xj = (ej, 0n, 0),Yj = (0n, ej, 0), j = 1, ··· , n and ej is the j’th canonical basis vector n of R , with the non trivial brackets

[Xi,Yj] = δi,jZ.

2 2.1 The unitary dual of Hn.

The unitary dual Hbn of Hn can be described as follows.

2.1.1 The inﬁnite dimensional irreducible representations

∗ 2 n For every λ ∈ R , there exists a unitary representation πλ of Hn on the Hilbert space L (R ), which is given by the formula

−2πiλt−2πi λ x·y+2πiλs·y n 2 n πλ(x, y, t)ξ(s) := e 2 ξ(s − x), s ∈ R , ξ ∈ L (R ), (x, y, t) ∈ Hn.

It is easily seen that πλ is in fact irreducible and that πλ is equivalent to πν if and only if λ = ν. Hn The representation πλ is equivalent to the induced representation τλ := indP χλ, where n P = {0n} × R × R is a polarization at the linear functional `λ((x, y, t)) := λt, (x, y, t) ∈ g −2πiλt and where χλ is the character of P deﬁned by χλ(0n, y, t) = e . The theorem of Stone-Von Neumann tells us that every inﬁnite dimensional unitary representation of Hn is equivalent to one of the πλ’s. (see [CG]).

2.1.2 The ﬁnite dimensional irreducible representations

Since Hn is nilpotent, every irreducible ﬁnite dimensional representation of Hn is one- dimensional, by Lie’s theorem. n n Any one-dimensional representation is a unitary character χa,b,(a, b) ∈ R ×R , of Hn, which is given by −2πi(a·x+b·y) χa,b(x, y, t) = e , (x, y, t) ∈ Hn. 1 For f ∈ L (Hn), let Z ˆ −2πi(x·a+y·b) n f(a, b) := χa,b(f) = f(x, y, t)e dxdydt, a, b ∈ R , Hn and ˆ kfk∞,0 := sup |χa,b(f)| = kfk∞. n a,b∈R

∗ 2.2 The topology of C\(Hn) ∗ ∗ ∗ Let C (Hn) denote the full C -algebra of Hn. We recall that C (Hn) is obtained by the 1 completion of L (Hn) with respect to the norm Z ∗ kfkC (Hn) = sup k f(x, y, t)π(x, y, t)dxdydtkop, where the supremum is taken over all unitary representations π of Hn. Deﬁnition 2.1. Let Hn ρ = indZ 1 2 be the left regular representation of Hn on the Hilbert space L (Hn/Z). Then the image ∗ ∗ 2n ρ(C (Hn)) is just the C -algebra of R considered as an algebra of convolution operators on

3 2 2n ∗ 2n L (R ) and ρ(C (Hn)) is isomorphic to the algebra C0(R ) of continuous functions vanishing 2n 1 ˆ at infinity on R via the Fourier transform. For f ∈ L (Hn) we have ρd(f)(a, b) = f(a, b, 0), n a, b ∈ R . ∗ Definition 2.2. Define for C (Hn) the Fourier transform F (c) of c by

2 n ∗ F (c)(λ) := πλ(c) ∈ B(L (R )), λ ∈ R and ∗ 2n F (c)(0) := ρ(c) ∈ C (R ).

∗ 2.2.1 Behavior on R

As for the topology of the dual space, it is well known that [πλ] tends to [πν] in Hbn if and ∗ only if λ tends to ν in R , where [π] denotes the unitary equivalence class of the unitary representation π. Furthermore, if λ tends to 0, then the representations πλ converge in the n dual space topology to all the characters χa,b, a, b ∈ R . 1 2 n n Let us compute for f ∈ L (Hn) the operator πλ(f). We have for ξ ∈ L (R ) and s ∈ R that Z πλ(f)ξ(s) = f(x, y, t)πλ(x, y, t)ξ(s)dxdydt n n R ×R ×R Z −2πiλt− 2πiλ x·y+2πiλs·y = f(x, y, t)e 2 ξ(s − x)dxdydt n n R ×R ×R Z −2πiλt− 2πiλ (s−x)·y+2πiλs·y (2.1) = f(s − x, y, t)e 2 ξ(x)dxdydt n n R ×R ×R Z λ = fˆ2,3(s − x, − (s + x), λ)ξ(x)dx. n 2 R

Here Z 2,3 −2πi(y·u+λt) n n fˆ (s, u, λ) = f(s, y, t)e dydt, (s, u, λ) ∈ R × R × R n R ×R denotes the partial Fourier transform of f in the variables y and t. Hence πλ(f) is a kernel operator with kernel λ (2.2) f (s, x) := fˆ2,3(s − x, − (s + x), λ), s, x ∈ n. λ 2 R

If we take now a Schwartz-functions f ∈ S(Hn), then the operator πλ(f) is Hilbert-Schmidt and its Hilbert-Schmidt norm kπλ(f)kH.S. is given by Z Z 2 2 ˆ2,3 2 (2.3) kπλ(f)kH.S. = |fλ(s, x)| dxds = |f (s, λx, λ)| dsdx < ∞. 2 2 R R ∗ ∗ Proposition 2.3. For any c ∈ C (Hn) and λ∈ R , the operator πλ(c) is compact , the ∗ 2 n mapping R → B(L (R )) : λ 7→ πλ(c) is norm continuous and tending to 0 for λ going to inﬁnity.

4 Proof. Indeed, for f ∈ S(Hn), the compactness of the operator πλ(f) is a consequence of (2.3) and by (2.1) we have the estimate: Z 2 ˆ2,3 λ ˆ2,3 ν 2 kπλ(f) − πν(f)kH.S = |f (s − x, − (s + x), λ) − f (s − x, − (s + x), ν)| dsdx n n 2 2 R ×R Z = |fˆ2,3(s, λx, λ) − fˆ2,3(s, νx, ν)|2dsdx n n R ×R Hence, since f is a Schwartz function, this expression goes to 0 if λ tends to ν by Lebesgue’s theorem of dominated convergence. Therefore the mapping λ 7→ πλ(f) is norm continuous. Furthermore, the Hilbert-Schmidt norms of the operators πλ(f) go to 0, when λ tends to ∗ inﬁnity. The proposition follows from the density of S(Hn) in C (Hn).

2.2.2 Behavior at 0

Let us now see the behavior of πλ(f) for Schwartz functions f ∈ S(Hn), as λ tends to 0. n 2 n n Choose a Schwartz-function η in S(R ) with L -norm equal to 1. For u = (a, b) in R ×R , λ ∈ ∗ R , we deﬁne the function η(λ, a, b) by b (2.4) η(λ, a, b)(s) := |λ|n/4e2πia·sη(|λ|1/2(s + )) s ∈ n. λ R n/4 1/2 n and let ηλ(s) = |λ| η(|λ| s), s ∈ R . Let us compute 0 cλ,u,u0 (x, y, t) = hπλ(x, y, t)η(λ, u), η(λ, u )i Z = e−2πiλt−2πi(λ/2)x·ye2πiλs·yη(λ, u)(s − x)η(λ, u0)(s)ds n R Z 0 = |λ|n/2e−2πiλt−2πi(λ/2)x·y e2πiλs·y−2πia·xe2πi(a−a )·s n R b b0 (2.5) η(|λ|1/2(s − x + ))η(|λ|1/2(s + ))ds λ λ Z n/2 −2πiλt−2πi(λ/2)x·y −2πib·y −2πia·x 2πiλs·y 2πi(a−a0)·(s− b ) = |λ| e e e e e λ n R b0 − b η(|λ|1/2(s − x)η(|λ|1/2(s + ))ds. λ Hence for u = u0 we get Z −2πiλt−2πi λ x·y −2πia·x−2πib·y 2πi(signλ)|λ|1/2s·y 1/2 cλ,u,u(x, y, t) = e 2 e e η(s − |λ| x)η(s)ds n Z R → e−2πia·x−2πib·y η(s)η(s)ds = e−2πia·x−2πib·y. n R It follows also that the convergence of the coeﬃcients cλ,u,u to the characters χa,b is uniform in u and uniform on compacta in (x, y, t) since Z −2πiλt−2πi λ x·y 2πi(signλ)|λ|1/2s·y |cλ,u,u(x, y, t) − χa,b(x, y, t)| = | (e 2 e n R η(s − |λ|1/2x)η(s) − |η(s)|2)ds| → 0 as λ → 0.

5 n n ∗ Proposition 2.4. For every u = (a, b) ∈ R × R , c ∈ C (Hn), we have that

lim kF (c)(λ)η(λ, u) − F\(c)(0)(u)η(λ, u)k2 = 0 λ→0 uniformly in (a, b).

∗ Proof. For c ∈ C (Hn) we have that 2 2 kF (c)(λ)η(λ, u) − F\(c)(0)(u)η(λ, u)k2 = kπλ(c)η(λ, u) − χa,b(c)η(λ, u)k2 =

= hπλ(c)η(λ, u) − χa,b(c)η(λ, u), πλ(c)η(λ, u) − χa,b(c)η(λ, u)i ∗ = hπλ(c ∗ c)η(λ, u), η(λ, u)i − χa,b(c)hπλ(c)η(λ, u), η(λ, u)i 2 − χa,b(c)hπλ(c)η(λ, u), η(λ, u)i + |χa,b(c)| 2 2 2 2 → |χa,b(c)| − |χa,b(c)| − |χa,b(c)| + |χa,b(c)| = 0.

2.3 A C∗-condition ∗ ∗ ∗ The aim of this section is to obtain a characterization of the C -algebra C (Hn) as a C - algebra of operator fields ([Lee1, Lee2]). ∗ Let us first define a larger C -algebra Fn.

Deﬁnition 2.5. Let Fn be the family consisting of all operator ﬁelds (F = F (λ))λ∈R satisfying the following conditions:

2 n ∗ 1. F (λ) is a compact operator on L (R ) for every λ ∈ R , ∗ 2n 2. F (0) ∈ C (R ), ∗ 2 n 3. the mapping R → B(L (R )) : λ 7→ F (λ) is norm continuous,

4. limλ→∞ kF (λ)kop = 0.

∗ Proposition 2.6. Fn is a C -algebra. Proof. The proof is straight forward.

∗ Proposition 2.7. The Fourier transform F : C (Hn) → Fn is an injective homomorphism. Proof. It is clear from the deﬁnition of F and Proposition 2.3 that F is a homomorphism ∗ with values in Fn. If F (c) = 0, then for each irreducible representation π of C (Hn), π(c) = 0. Hence c = 0.

n ∗ Lemma 2.8. Let ξ ∈ S(R ). Then, for any λ ∈ R , 1 Z ξ = n hξ, η(λ, u)iη(λ, u)du, |λ| n n R ×R 2 n where η(λ, u) is given by (2.4), the integral converging in L (R ).

6 n Proof. Let ξ ∈ S(R ). Then Z hξ, η(λ, a, b)iη(λ, a, b)(x)dadb n n R ×R Z Z −2πia·s b 2πia·x b = ξ(s)e ηλ(s + ))ds e ηλ(x + )dadb n n n λ λ R ×R R (by Fourier’s inversion formula) Z b b n = ξ(x)ηλ(x + )ηλ(x + )db = |λ| ξ(x) n n λ λ R ×R 1 R giving ξ = n n n hξ, η(λ, a, b)iη(λ, a, b)dadb. |λ| R ×R Furthermore, since ξ is a Schwartz function, it follows that the mapping

Z b (a, b) → hξ, η(λ, a, b)i = |λ|n/4 ξ(s)e−2πia·sη(|λ|1/2(s + ))ds n λ R is also a Schwartz function in the variables a, b. Hence the integral R n 2 n n n hξ, η(λ, a, b)iη(λ, a, b)dadb converges in S( ) and hence also in L ( ). R ×R R R

Remark 2.9. By Lemma 2.8, 1 Z 1 Z πλ(f)ξ = n πλ(f)η(λ, u)hξ, η(λ, u)idu = n πλ(f) ◦ Pη(λ,u)ξdu |λ| 2n |λ| 2n R R ∗ for any f ∈ C (Hn), where Pη(λ,u) is the orthogonal projection onto the one dimensional subspace Cη(λ, u). 2 n Deﬁnition 2.10. For a vector 0 6= η ∈ L (R ), we let Pη be the orthogonal projection onto the one dimensional subspace Cη. ∗ ∗ 2n Deﬁne for λ ∈ R and h ∈ C (R ) the linear operator Z ˆ du (2.6) νλ(h) := h(u)Pη(λ,u) n . 2n |λ| R Proposition 2.11.

∗ 2n 1. For every λ ∈ R and h ∈ S(R ) the integral (2.6) converges in operator norm.

2. ν (h) is compact and kν (h)k ≤ khk ∗ 2n . λ λ op C (R ) ∗ 2n ∗ ∗ ∗ 2n 3. The mapping νλ : C (R ) → Fn is involutive, i.e. νλ(h ) = νλ(h) , h ∈ C (R ), where ∗ 2n by νλ we denote also the extension of νλ to C (R ). 2 Proof. Since kPη(λ,u)kop = kη(λ, u)k2 = 1, we have that Z Z ˆ ˆ du ˆ du khk1 kνλ(h)kop = k h(u)Pη(λ,u) n kop ≤ |h(u)| n = n . 2n |λ| 2n |λ| |λ| R R R ˆ du 2n Hence the integral 2n h(u)Pη(λ,u) n converges in operator norm for h ∈ S( ). R |λ| R

2n Let h ∈ S(R ). Then hˆ = hˆ∗. This gives Z Z ∗ ˆ du ∗ ˆ du νλ(h) = ( h(u)Pη(λ,u) n ) = h(u)Pη(λ,u) n 2n |λ| 2n |λ| R R Z du ˆ∗ ∗ = h (u)Pη(λ,u) n = νλ(h ). 2n |λ| R

8 ∗ Theorem 2.12. Let a ∈ C (Hn) and let A be the operator ﬁeld A = F (a), i. e.

∗ ∗ 2n A(λ) = πλ(a), λ ∈ R ,A(0) = ρ(a) ∈ C (R ). Then lim kA(λ) − νλ(A(0))kop = 0. λ→0 2 n n Proof. Let f ∈ S(Hn), ξ ∈ L (R ), η ∈ S(R ), kηk2 = 1. Then by (2.1) and (2.7) Z ˆ2,3 λ ((πλ(f) − νλ(ρ(f))ξ)(x) = f (x − s, − (x + s), λ)ξ(s)ds n 2 ZR Z ˆ2,3 b b db − f (x − s, b, 0)ξ(s)ηλ(s + )ηλ(x + ) n ds n n λ λ |λ| R R Z Z ˆ2,3 λ = f (x − s, − (x + s), λ)ηλ(b)¯ηλ(b)ξ(s)dbds n n 2 ZR ZR ˆ2,3 − f (x − s, λ(b − x), 0)ξ(s)ηλ(s − x + b)ηλ(b)dbds. n n R R Let Z ˆ2,3 λ ˆ2,3 λ uλ(x, b) = ξ(s)ηλ(s − x + b)(f (x − s, − (x + s), λ) − f (x − s, − (x + s), 0))ds, n 2 2 ZR ˆ2,3 λ ˆ2,3 vλ(x, b) = ξ(s)ηλ(s − x + b)(f (x − s, − (x + s), 0) − f (x − s, λ(b − x), 0))ds n 2 R and Z Z ˆ2,3 λ wλ(x) = f (x − s, − (x + s), λ)ξ(s)ηλ(b)(ηλ(b) − ηλ(s − x + b))dbds. n n 2 R R We have Z Z (2.8) ((πλ(f) − νλ(ρ(f))ξ)(x) = uλ(x, b)ηλ(b)db + vλ(x, b)ηλ(b)db + wλ(x). n n R R

Thus to prove kπλ(f) − νλ(ρ(f))kop → 0 as λ → 0 it is enough to show that kuλk2 ≤ δλkξk2, kvλk2 ≤ ωλkξk2 and kwλk2 ≤ λkξk2, where δλ, ωλ, λ → 0 as λ → 0. We have Z 1 2,3 λ 2,3 λ 2,3 λ fˆ (x − s, − (x + s)), λ) − fˆ (x − s, − (x + s), 0) = λ ∂3fˆ (x − s, − (x + s)), tλ)dt 2 2 0 2 and λ 1 fˆ2,3(x − s, − (x + s)), 0) − fˆ2,3(x − s, λ(b − x), 0) = λ( (s − x) − (s − x + b)) 2 2 Z 1 2,3 1 × ∂2fˆ (x − s, λ(b − x) + t(λ( (s − x) − (s − x + b)), 0)dt. 0 2

Hence, since f ∈ S(Hn), there exists a constant C > 0 such that λ λ C |f 2,3(x − s, − (x + s)), λ) − fˆ2,3(x − s, − (x + s), 0)| ≤ |λ| , 2 2 (1 + kx − sk)2n+1

9 and λ |fˆ2,3(x − s, − (x + s)), 0) − fˆ2,3(x − s, λ(b − x), 0)| 2 C ≤ |λ|(ks − x + bk + ks − xk) (1 + kx − sk)4n+1 ∗ n for all λ ∈ R , x, s ∈ R . Therefore we see that Z 2 2 kuλk2 = |uλ(x, b)| dxdb n n R ×R Z Z C 2 ≤ |ξ(s)ηλ(s − x + b)||λ| 2n+1 ds dxdb n n n (1 + kx − sk) R ×R R Z 2 2 0 |ξ(s)| 2 ≤ |λ| C 2 |ηλ(s − x + b)| dbdxds 3n (1 + kx − sk) R 00 2 2 ≤ C |λ| kξk2. Similarly Z 2 2 kvλk2 = |vλ(x, b)| dxdb n n R ×R Z Z ≤ |ξ(s)ηλ(s − x + b)||λ|(ks − x + bk + ks − xk) n n n R ×R R C 2 ds dbdx (1 + kx − sk)4n+1 Z 0 2 2 2 ≤ C |ξ(s)ηλ(s − x + b)| |λ| (ks − x + bk + ks − xk) 3n R 1 dsdbdx (1 + kx − sk)4n+1 Z 0 n/4 1/2 2 2 2 dsdbdx ≤ 2C |ξ(s)|λ| η(|λ| (s − x + b))| |λ| ks − x + bk 4n+1 3n (1 + kx − sk) R Z 0 n/4 1/2 2 2 2 dsdbdx + 2C |ξ(s)|λ| η(|λ| (s − x + b))| |λ| ks − xk 4n+1 3n (1 + kx − sk) R Z 0 n/4 1/2 2 dsdbdx ≤ 2C |λ| |ξ(s)|λ| η˜(|λ| (s − x + b))| 4n+1 3n (1 + kx − sk) R Z 0 2 n/4 1/2 2 dsdbdx + 2C |λ| |ξ(s)|λ| η(|λ| (s − x + b))| 4n−1 3n (1 + kx − sk) R 00 2 2 2 ≤ C |λ|(kη˜k2 + |λ|kηk2)kξk2, 0 00 for some constants C ,C > 0, where the functionη ˜ is deﬁned byη ˜(s) := kskη(s), s ∈ R. n Since η ∈ S(R ), we can use the same arguments to see that Z 2 2 kwλk2 = |wλ(x)| dx n R Z Z Z 2 n/4+1/2 C ≤ |ξ(s)|ηλ(b)||λ| (ks − xk) 4n+1 dbds dx n n n (1 + kx − sk) R R R Z 2 0 n/2+1 2 2 kx − sk dsdxdb 00 n/2+1 2 2 ≤ C |λ| |ξ(s)| |ηλ(b)| 4n+1 ≤ C |λ| kξk2kηk2. 3n 1 + kx − sk R

10 We have proved therefore kπλ(f) − νλ(ρ(f))k → 0 as λ → 0 for f ∈ S(Hn). Since S(Hn) is ∗ ∗ dense in C (Hn), the statement holds for any a ∈ C (Hn).

n ∗ 2n Deﬁnition 2.13. For η ∈ S(R ) we deﬁne the linear mapping νη := ν : C (R ) → Fn by ∗ ν(h)(λ) = νλ(h), λ ∈ R and ν(h)(0) = h.

∗ 2n Proposition 2.14. The mapping ν : C (R ) → Fn has the following properties: 1. kνk = 1.

0 ∗ 2n 2. For every h, h ∈ C (R ), we have that 0 0 lim kνλ(h · h ) − νλ(h) ◦ νλ(h )kop = 0 λ→0 and also ∗ ∗ lim kνλ(h ) − νλ(h) kop = 0. λ→0

2n ∗ 2n 3. For (a, b) ∈ R and h ∈ C (R ) we have that

lim kν(h)(λ)η(λ, a, b) − hˆ(a, b)η(λ, a, b)k2 = 0. λ→0

ˆ 4. limλ→0 kν(h)(λ)k = khk∞. Proof. (1) follows from Proposition 2.11. 0 2n 0 To prove (2) we take for h, h ∈ S(R ) two elements f, f ∈ S(Hn), such that ρ(f) = h, ρ(f 0) = h0. Then ρ(f ∗ f 0) = h · h0 and

0 0 0 0 kνλ(h · h ) − νλ(h) ◦ νλ(h )kop ≤ kνλ(h · h ) − πλ(f ∗ f )kop 0 0 + kνλ(h) ◦ νλ(h ) − πλ(f) ◦ πλ(f )kop 0 0 ≤ kνλ(h · h ) − πλ(f ∗ f )kop 0 ∗ (2.9) + kf kC (Hn)kνλ(h) − πλ(f)kop 0 0 + khk ∗ 2n kν (h ) − π (f )k . C (R ) λ λ op 0 0 Hence, by Theorem 2.12, limλ→0 kνλ(f ∗ f ) − νλ(f) ◦ νλ(f )kop = 0. Furthermore

∗ ∗ ∗ ∗ ∗ ∗ kνλ(h ) − νλ(h) kop ≤ kνλ(h ) − πλ(f )kop + kνλ(h) − πλ(f) kop → 0 as λ → 0.

We conclude by the usual approximation argument. ∗ 2n For assertion (3), using Propositions 2.4 and Theorem 2.12, it suﬃces to take for h ∈ C (R ) ∗ an element c ∈ C (Hn), for which ρ(c) = h. The last statement follows from Proposition 2.11 and assertion (3).

Deﬁnition 2.15. Let Dν(Hn) be the subspace of the algebra Fn, consisting of all the ﬁelds

(F (λ))λ∈R ∈ Fn, such that

lim kF (λ) − νλ(F (0))kop = 0. λ→0

11 ∗ Our main theorem of this section is the following characterisation of C (Hn).

∗ ∗ Theorem 2.16. The Heisenberg C -algebra C (Hn) is isomorphic to Dν(Hn).

0 Proof. First we show that Dν(Hn) is a ∗-subalgebra of Fn. Indeed if F,F ∈ Dν(Hn), then

0 0 0 0 kνλ(F (0) + F (0)) − πλ(F + F )kop ≤ kνλ(F (0)) − πλ(F )kop + kνλ(F (0)) − πλ(F )kop → 0 as λ → 0.

0 0 and since limλ→0 kνλ(F · F (0)) − νλ(F (0)) ◦ ν(F (0))kop = 0 it follows that

0 0 kνλ(F (0) · F (0)) − πλ(F · F )kop → 0.

Proposition 2.14 tells us that Dν(Hn) is also invariant under the involution ∗. In order to see that Dν(Hn) is closed, let F ∈ Fn be contained in the closure of Dν(Hn). Let 0 0 0 ε > 0. Choose F ∈ D (H ), such that kF −F k < ε. In particular, kF (0)−F (0)k ∗ 2 < ε. ν n Fn C (R ) Thus there exists λ0 > 0, such that

0 0 kπλ(F ) − νλ(F (0))kop < ε for all |λ| < |λ0|, whence

0 0 0 0 kπλ(F ) − νλ(F (0))kop = kπλ(F ) − πλ(F ) + πλ(F ) − νλ(F (0)) + νλ(F (0)) − νλ(F (0))kop

≤ 3ε, for |λ| < |λ0|.

∗ Hence Dν(Hn) is a C -subalgebra of Fn. Let I0 := {F ∈ Fn,F (0) = 0} and let I00 = {F ∈ I0; limλ→0 kF (λ)kop = 0}. Then I0 and I00 are closed two sided ideals of Fn and it follows from the deﬁnition of Fn that I00 is just ∗ the algebra C0(R , K). It is clear that Dν(Hn) ∩ I0 = I00. But Dν(Hn) ∩ I0 is the kernel in ∗ 2n Dν(Hn) of the homomorphism δ0 : Fn → C (R ); F 7→ F (0). ∗ 2n Since im(ν) ⊂ Dν(Hn), the canonical projection Dν(Hn) → C (R ): F 7→ F (0) is surjective ∗ 2n and has the ideal I00 as its kernel. Thus Dν(Hn)/I00 = C (R ) and therefore Dν(Hn) is an ∗ 2n extension of I00 by C (R ). Moreover,

Dν(Hn) = I00 + im(ν).

Since for every irreducible representation π of Dν(Hn), we have either π(I00) 6= 0, and then ∗ ∗ 2n π = πλ for some λ ∈ R or π = 0 on I00 and then π must be a character of C (R ). Hence Dˆν(Hn) = Hˆn as sets. That topologies of these spaces agree follows from the equality

ˆ ∗ 2n lim kτ(h)(λ)kop = khk∞, ∀h ∈ C (R ), λ→0 which is due to Proposition 2.14. ∗ ∗ By Theorem 2.12, F (S(Hn)) ⊂ Dν(Hn). Hence the C - algebra C (Hn) can be injected into Dν(Hn). ∗ Since Dν(Hn) is a type I algebra and the dual spaces of Dν(Hn) and of C (Hn) are the same, ∗ we have that F (C (Hn)) is equal to Dν(Hn) by the Stone -Weierstrass theorem (see [Di]).

12 ∗ ∗ Remark 2.17. Another characterisation of the C -algebra C (Hn) is given (without proof) ∗ in a short paper by Gorbachev [Gor]. For n = 1 and λ ∈ R he deﬁnes an operator-valued 2 measure µλ on R given on the product of two intervals [s, t] × [e, d] by µλ([s, t] × [e, d]) = e , d s,t −1 s,t P λ λ FP F , where P is the multiplication operator by the characteristic function of 2 2 2 ∗ [t, s] on L (R) and F is the Fourier transform on L (R). For f ∈ C0(R ), λ ∈ R let Z y(f)(λ) = f(a, b)dµλ(a, b) 2 R ∗ ∗ and y(f)(0) = f. Gorbachev states that C (H1) is isomorphic to the C -algebra of operator ∗ 2 ∗ ﬁelds B = {B(λ) = y(f)(λ) + a, λ ∈ R ,B(0) = f, f ∈ C0(R ), a ∈ C0(R , K)}.

2.4 Almost homomorphisms and Heisenberg property ∗ 2n Deﬁnition 2.18. A bounded mapping τ : C (R ) → Fn is called an almost homomorphism if

lim kτλ(αh + βf) − ατλ(h) − βτλ(f)kop = 0, λ→0 0 0 lim kτλ(h · h ) − τλ(h) ◦ τλ(h )kop = 0, λ→0 ∗ ∗ ∗ 2n lim kτλ(h ) − τλ(h) kop = 0, α, β ∈ C, f, h ∈ C (R ). λ→0 The mapping ν from the previous section is an example of such almost homomorphism. ∗ 2n Let τ be an arbitrary almost homomorphism such that τ(f)(0) = f for any f ∈ C (R ). We deﬁne as before Dτ (Hn) to be the subspace of the algebra Fn, consisting of all the ﬁelds

F = (F (λ))λ∈R ∈ Fn, such that

lim kF (λ) − τλ(F (0))kop = 0. λ→0 Using the same arguments as the one in the proof of Theorem 2.16 one can easily prove the following

∗ ∗ Proposition 2.19. The subspace Dτ (Hn) of the C -algebra Fn is itself a C -algebra. The ∗ ∗ 2n ∗ algebra Dτ (Hn) is an extension of C0(R , K) by C (R ), i.e., C0(R , K) is a closed ∗-ideal ∗ ∗ 2n in Dτ (Hn) such that Dτ (Hn)/C0(R , K) is isomorphic to C (R ). ∗ 2n Deﬁnition 2.20. We say that an almost homomorphism τ : C (R ) → Fn has the Heisen- ∗ ∗ berg property, if the C -algebra Dτ (Hn) is isomorphic to C (Hn).

Remark 2.21. As for the mapping ν we have that the dual spaces of Dτ (Hn) and of Hn coincides as sets. The necessary and the suﬃcient conditions for them to coincide as topological spaces is ˆ ∗ n lim kτλ(h)kop = khk∞, h ∈ C (R ). λ Remark 2.22. Using the notion of Busby invariant for a C∗-algebra extension and the ∗ ∗ 2n pullback algebra ([W]), one can show that any extension B ⊂ Fn of C0(R , K) by C (R ) is isomorphic to Dτ (Hn) for some almost homomorphism τ. The Busby invariant of such ∗ 2n ∗ ∗ ∗ extension is b : C (R ) → Cb(R ,B(H))/C0(R , K), b(h) = τ(h) + C0(R , K).

13 Question. ∗ ∗ What mappings τ give us C -algebras Dτ (Hn), which are isomorphic to C (Hn)? ∗ Using a procedure described in [De] one can construct families of C -algebras of type Dτ (Hn) ∗ which are isomorphic to Dν(Hn) and therefore to C (Hn). ∗ Next example shows that there is no topological obstacle for a C -algebra of type Dτ (Hn) ∗ ∗ to be non-isomorphic to C (Hn). Namely, there is a C -algebras Dτ (Hn) with the spectrum ∗ equal to Hbn and such that Dτ (Hn) 6' C (Hn). We recall ﬁrst that if A, C are C∗-algebras, then an extension of C by A is a short exact sequence

β (2.10) 0 → A →α B → C → 0 of C∗-algebras. One says that the exact sequence splits if there is a cross-section ∗- homomorphism s : C → B such that β ◦ s = IC. It is known that the extension ∗ ∗ ∗ 2n (2.11) 0 → C0(R , K) → C (Hn) → C (R ) → 0 does not split (see [R] and references therein) while there exists a large number of splitting ∗ extensions B and therefore non-isomorphic to C (Hn) such that Bˆ = Hˆn (see [De, VII.3.4]). Here is a concrete example inspired by [De]. 2 n Example 2.23. Let {ξ } 2n be an orthonormal basis of the Hilbert space L ( ). Let Z Z∈Z R 2n PZ ,Z ∈ Z , be the orthogonal projection onto the one-dimensional CξZ . We define a homo- ∗ 2n morphism ν from C (R ) to Fn by X 1/2 ∗ ∗ 2n ν(ϕ)(λ) := ϕˆ(|λ| Z)PZ , λ ∈ R , ν(ϕ)(0) := ϕ, ϕ ∈ C (R ). 2n Z∈Z 2n 2n We note that since for each λ 6= 0 and each compact subset K ⊂ R , the set {Z ∈ Z : 1/2 2n |λ| ∈ K} is finite and sinceϕ ˆ ∈ C0(R ), one can easily see that ν(ϕ)(λ) is compact. Moreover 1/2 kν(ϕ)(λ)kop = sup |ϕ(|λ| Z)|. 2n Z∈Z 2n ∗ 2n Since we can find for every vector u ∈ R and λ ∈ R a vector Zλ ∈ Z , such that 1/2 limλ→0 |λ| Zλ = u, we see that

(2.12) lim kνλ(ϕ)kop = kϕˆk∞ = kϕk ∗( 2n). λ→0 C R

∗ 3 The C -algebra of the thread-like Lie groups GN

For N ≥ 3, let gN be the N-dimensional real nilpotent Lie algebra with basis X1,...,XN and non-trivial Lie brackets

[XN ,XN−1] = XN−2,..., [XN ,X2] = X1.

The Lie algebra gN is (N − 1)-step nilpotent and is a semi-direct product of RXN with the abelian ideal N−1 X (3.13) b := RXj. j=1

14 Let bj := span{Xi, i = 1, ··· , j}, 1 ≤ j ≤ N − 1.

Note that g3 is the three dimensional Heisenberg Lie algebra. Let GN := exp(gN ) be the associated connected, simply connected Lie group. Let also Bj := exp(bj) and B := exp(b). Then for 3 ≤ M ≤ N we have GM ' GN /BN−M .

3.1 The unitary dual of GN

In this section we describe the unitary irreducible representations of GN up to a unitary equivalence. PN−1 ∗ ∗ For ξ = j=1 ξjXj ∈ gN , the coadjoint action is given by

N−1 ∗ X ∗ Ad (exp(−tXN))ξ = pj(ξ, t)Xj , j=1

where, for 1 ≤ j ≤ N − 1, pj(ξ, t) is a polynomial in t deﬁned by

j−1 X tk p (ξ, t) = ξ . j k! j−k k=0

∗ Moreover, if ξj 6= 0 for at least one 1 ≤ j ≤ N − 2, then Ad (GN)ξ is of dimension two, and ∗ ∗ ∗ ∗ N Ad (GN)ξ = { Ad (exp(tXN))ξ + RXN, t ∈ R}. We shall always identify gN with R via the PN ∗ ∗ mapping (ξN , . . . , ξ1) → j=1 ξjXj and the subspace V = {ξ ∈ gN : ξN = 0} with the dual space of b. For ξ ∈ V and t ∈ R, let

∗ t · ξ = Ad (exp(tXN))ξ

1 (3.14) = 0, ξ − tξ + ... + (−t)N−2ξ , . . . , ξ − tξ , ξ . N−1 N−2 (N − 2)! 1 2 1 1

As in [AKLSS], we deﬁne the function ξb on R by

1 N−2 (3.15) ξb(t) := (t · ξ)N−1 = ξN−1 − tξN−2 + ... + (−t) ξ1. (N − 2)!

Then the mapping ξ → ξbis a linear isomorphism of V onto PN−2, the space of real polynomials of degree at most N − 2. In particular, ξk → ξ coordinate-wise in V as k → ∞ if and only ∗ if ξbk(t) → ξb(t) for all t ∈ R. Also, the mapping ξ → ξb intertwines the Ad -action and translation in the following way:

td· ξ(s) = (s · (t · ξ))N−1

= ((s + t) · ξ)N−1 = ξb(s + t) for ξ ∈ V and s, t ∈ R. By Kirillov’s orbit picture of the dual space of a nilpotent Lie group, we can describe the irreducible unitary representations of GN in the following way (see [CG] for details). For any

15 ˆ G non-constant polynomial p = ` ∈ PN−2 we consider the induced representation π` = indBχ`, where χ` denotes the unitary character of the abelian group B deﬁned by: −2πih`,Ui χ`(exp(U)) = e ,U ∈ b.

Since b is abelian of codimension 1, it is a polarization at ` and so π` is irreducible. Every inﬁnite dimensional irreducible unitary representation of GN arises in this manner up to equivalence.

∗ Let us describe the representation π`, ` ∈ b , explicitly. The Hilbert space H` of the repre- 2 ˜ sentation π` is the space L (GN /B, χ`) consisting of all measurable functions ξ : GN → C, ˜ −1 ˜ such that ξ(gb) = χ`(b )ξ(g) for all b ∈ B and all g ∈ G outside some set of measure of ˜ 2 Lebesgue measure 0 and such that the function |ξ| is contained in L (GN /B). We can identify 2 2 ˜ the space L (GN /B, χ`) in an obvious way with L (R) via the isomorphism U : ξ 7→ ξ where ˜ −1 2 ξ(exp(sXN )b) := χ`(b )ξ(s), s ∈ R, b ∈ B. Hence for g = exp(tXN )b and ξ ∈ L (R) we have an explicit expression for the operator π`(g): ˜ −1 π`(g)ξ(s) = ξ(g exp(sXN ))(3.16) ˜ −1 = ξ(b exp((s − t)XN )) ˜ −1 = ξ(exp((s − t)XN )(exp((t − s)XN )b exp((s − t)XN )))

= χ`(exp((t − s)XN )bexp((s − t)XN ))ξ(s − t) −2πi`(Ad(exp((t−s)X ) log(b)) = e N ξ(s − t), s ∈ R. ∗ We can parametrize the orbit space gN /GN in the following way. First we have a decompo- sition N−2 ∗ [ ∗ j [ ∗ gN /GN = gN /GN X , j=1 where

∗ j ∗ gN := {` ∈ gN , `(Xi) = 0, i = 1, ··· , j − 1, `(Xj) 6= 0} and where

∗ ∗ X := {` ∈ gN , `(Xj) = 0, j = 1, ··· ,N − 2} denotes the characters of GN . A character of the group GN can be written as χa,b, a, b ∈ R, where

−2πiaxN −2πibxN−1 χa,b(xN , xN−1,, ··· , x1) := e , (xN ,, ··· , x1) ∈ GN . ∗ j For any ` ∈ gN ,N − 2 ≥ j ≥ 1 there exists exactly one element `0 in the GN -orbit of `, which satisﬁes the conditions

`0(Xj) 6= 0, `0(Xj+1) = 0, `0(XN ) = 0.

∗ We can thus parametrize the orbit space gN /GN , and hence also the dual space GdN , with the sets N−2 [ j [ ∗ SN := SN X , j=1

16 j ∗j ∗ j where SN := SN ∩ g = {` ∈ gN , `(Xk) = 0, k = 1, ··· , j − 1, j + 1, `(Xj) 6= 0}. Let

N−2 gen [ j SN := SN j=1

be the family of points in SN , whose GN -orbits are of dimension 2.

3.2 The topology of GdN

The topology of the dual space of GN has been studied in detail in the papers [ALS] and [AKLSS] based on the methods developed in [LRS] and [L]. We need the following description of the convergence of sequences (πk)k of representations in GdN . Let (πk)k be a sequence in GdN . It is said to be properly convergent if it is convergent and all cluster points are limits. It is known (see [LRS]) that any convergent sequence has a properly convergent subsequence.

gen Proposition 3.1. Suppose that (πk = π`k )k, (`k ∈ SN , k ∈ N) is a sequence in GdN that has a cluster point. Then there exists a subsequence, (also indexed by the symbol k for simplicity), called with perfect data such that (πk)k is properly converging and such that the polynomials pk, k ∈ N, associated to πk have the following properties: The polynomials pk have all the same degree d. Write

d Y k ˆ pk(t) := ck (t − aj ) = `k(t), t ∈ R, `k ∈ V. j=1

k There exist 0 < m ≤ 2d, real sequences (ti )k and polynomials qi of degree di ≤ d, i = 1, ··· , m, such that

k k i 1. limk→∞ pk(t + ti ) → qi(t), t ∈ R, 1 ≤ i ≤ m or equivalently limk→∞ ti · `k → ` , where i i ` in V such that `ˆ (t) = qi(t).

k k 0 2. limk→∞ |ti − ti0 | = +∞, for all i 6= i ∈ {1, ··· , m}. 3. If C = {i ∈ {1, ··· , m, }, `i is a character } then for all i ∈ C

k k (a) limk→∞ |ti − aj | = +∞ for all j ∈ {1, ··· , d}; k k k k (b) there exists an index j(i) ∈ {1, ··· , d} such that |ti − aj(i)| ≤ |ti − aj | for all j ∈ {1, ··· , d}; let k k k ρi := |ti − aj(i)|;

k k |ti −aj | (c) there exists a subset L(i) ⊂ {1, ··· , m}, such that limk→∞ k exists in R for ρi k k |ti −aj | every j ∈ L(i) and such that limk→∞ k = +∞ for j 6∈ L(i); ρi k k (d) the polynomials (ti +sρi )·pk in t converge uniformly on compacta to the constants

k k i lim (ti + sρi ) · pk(t) = p (s), s ∈ R, k→∞ and these constants deﬁne a real polynomial of degree #L(i) in s.

17 (e) If i0 6= i ∈ C, then L(i) ∩ L(i0) = ∅.

k 4. Let D = {1, ··· , m}\ C and write ρi := 1 for i ∈ D. For i ∈ D, let

k k J(i) := {1 ≤ j ≤ d, lim |ti − aj | = ∞}. k→∞

∗ Suppose that (tk)k is a real sequence, such that limk→∞ tk · `k → ` in gN , then k (a) if ` is a non-character, then the sequence (|tk − ti |)k is bounded for some i∈ / C; t −ak (b) if ` is a character, then lim k j exists for some i ∈ C and some j ∈ L(i) k→∞ tk−ak i j ∗ and `|b = qi(s)XN−1 for some s ∈ R.

m [ k k k k Sk := ( [ti − skρi , ti + skρi ]); Tk := R \ Sk, k ∈ N. i=1

Then for any sequence (tk)k, tk ∈ Tk, we have tk · lk → ∞.

We say that the sequence (sk)k is adapted to the sequence (`k).

Proof. We may assume that (πk)k is properly convergent with limit set L. We can also assume, by passing to a subsequence, that each pk has degree d. By [L] the number of non-characters in L is finite. Let this subset of non-characters be denoted by Lgen. If Lgen is non-empty by passing further to a subsequence we may assume the sequence (πk)k converges iσ-times to each non-character σ ∈ Lgen (see p.34, [AKLSS] for the definition of m-convergence and P p.253 [ALS]). Let s = σ∈Lgen iσ. Then there exist non-constant polynomials q1, . . . , qs of k k degree di ≤ d, i = 1, . . . , s, and sequences (t1)k,..., (ts )k such that the conditions (1) and gen (2) are fulfilled and for each σ ∈ L there are iσ equal polynomials amongst q1, . . . , qs corresponding to σ. Then if (tk)k is a real sequence such that tk · `k → `, ` is a non-character, gen then ` corresponds to some σ ∈ L and we may assume that `ˆ= qi for some i ∈ 1, . . . , s. It k follows from the definition of iσ-convergence that the sequences (tk · `k) and (ti · `k) are not k disjoint implying |tk − ti | is bounded and therefore (4a). If (πk)k has a character as a limit point then passing if necessary to a subsequence we can k find a maximal family of real sequences (tl )k, l < s ≤ m ≤ d, constant polynomials ql, non- k negative sequences (ρl )k and polynomials pl satisfying (1) − (4) (see Definition 6.4 and the discussion before in [ALS]). k The condition (4b) follows from the maximality of the family of sequences (tl )k and the proof of Proposition 6.2, [ALS]. Suppose now that we have a sequence (tk)k such that tk ∈ Tk for every k and such that some subsequence (also indexed by k for simplicity of notations) (tk · `k)k converges to an ∗ k ` ∈ g . By condition (5) then either for some i ∈ D, the sequence (tk − ti )k is bounded, i.e k k k k tk ∈ [ti − skρi , ti + skρi ] for k large enough, which is impossible, or we have an i ∈ C, such

18 t −ak that lim k j exists for some j ∈ L(i). But then k→∞ tk−ak i j

k |tk−ti | k k k k and so the sequence ( k )k is bounded, i.e tk ∈ [ti − skρi , ti + skρi ] ⊂ Sk for k large ρi enough, a contradiction. Hence limk tk · `k = ∞ whenever tk ∈ Tk for large k.

1 2 1 Example 3.2. Let us consider the Heisenberg group G3. Then S3 = S3 ∪ R . Let (`k) ∈ S3 . ∗ ∗ Then `k = λkX1 , λk ∈ R . The associated polynomials are pk(t) = −λkt (d = 1, ck = −λk, k a1 = 1). Assume that (`k) is a sequence with perfect data. Then either π`k converges to π`, 1 ` ∈ S3 , or π`k converges to a character and in this case λk → 0 as k → ∞. We shall consider 1 ∗ now the second case. So we have m = 1 and ` = X2 with the corresponding polynomial k k 1 q1(t) = 1 and t1 = −1/λk and thus ρ1 = 1/|λk|. The polynomial p (s) is the limit

k k lim pk(t1 + sρi + t) = lim (−λk)(−1/λk + s/|λk| + t) = lim (1 − signλks − λkt). k→∞ k→∞ k→∞

Since (`k) is a sequence with perfect data, the sign of λk is constant, implying q11(s) = 1+s, where = ±1. A real sequence (sk) is adapted to (`k) if and only if sk → ∞ and sk|λk| → 0.

3.3 A C∗-condition ∗ ∗ Let C (GN ) be the full C -algebra of GN that is the completion of the convolution algebra 1 L (GN ) with respect to the norm Z ∗ kfkC (GN ) = sup k f(g)π`(g)dgkop. `∈SN GN

1 ˆ2 ∗ Deﬁnition 3.3. Let f ∈ L (GN ). Deﬁne the function f on R × b by Z 2 −2πi`(log(u)) ∗ fˆ (s, `) := f(s, u)e du, s ∈ R, ` ∈ b . B

1 1 ˆ2 We denote by Lc (GN ) the space of functions f ∈ L (GN ), for which f is contained in ∞ ∗ ∞ ∗ 1 Cc (R×b ), the space of compactly supported C -functions on R×b . The subspace Lc (GN ) 1 ∗ ∗ is dense in L (GN ) and hence in the full C -algebra C (GN ) of GN .

1 gen Proposition 3.4. Take f ∈ Lc (GN ) and let ` ∈ SN . Then the operator π`(f) is a kernel operator with kernel function

ˆ2 f`(s, t) = f (s − t, t · `), s, t ∈ R.

19 2 Proof. Indeed, for ξ ∈ L (R), s ∈ R, we have that Z π`(f)ξ(s) = f(g)π`(g)ξdg GN Z Z = f(t, b)e−2πi`(Ad(exp((t−s)XN) log(b))ξ(s − t)dbdt( by 3.16) R B Z Z ∗ (3.17) = f(s − t, b)e−2πi(Ad (exp(tXN )(`)(log(b))ξ(t)dbdt B ZR ˆ2 ∗ = f (s − t, Ad (exp(tXN (`))ξ(t)dt ZR = fˆ2(s − t, t · `)ξ(t)dt. R

Definition 3.5. Let c := span {X1,, ··· ,XN−2}. Then c is an abelian ideal of gN , the 2 algebra gN /c is abelian and isomorphic to R and C := exp(c) is an abelian closed normal subgroup of GN . Let GN ρ = indC 1 2 be the left regular representation of GN on the Hilbert space L (GN /C). Then the image ∗ ∗ 2 2 2 ρ(C (GN )) is the C -algebra of R considered as algebra of convolution operators on L (R ) ∗ 2 and hence ρ(C (GN )) is isomorphic to the algebra C0(R ) of continuous functions vanishing 2 1 at infinity on R . As for the Heisenberg algebra we have that if f ∈ L (GN ) then the Fourier ∗ 2 transform ρd(f)(a, b) of ρ(f) ∈ C (R ) equals fˆ(a, b, 0,..., 0). ∗ ∗ ∗ Our aim is to realize the C -algebra C (GN ) as a C -algebra of operator fields.

∗ Definition 3.6. For a ∈ C (GN ) we define the Fourier transform F (a) of a as operator field

gen ∗ 2 F (a) := {(A(`) := π`(a), ` ∈ SN ,A(0) := ρ(a) ∈ C (R )}.

j Remark 3.7. We observe that the spaces SN , j = 1, ··· ,N − 2, are Hausdorﬀ spaces if we j equip them with the topology of GdN . Indeed, let (`k)k be a sequence in SN , such that the j sequence of representations (π`k )k converges to some π` with ` ∈ SN . Then the numerical sequence (λk := `k(Xk))k converges to λ := `(Xj) 6= 0. Suppose now that the same sequence 0 (π`k ) converges to some other point π` . Then there exists a numerical sequence (tk)k such that ∗ 0 ∗ k→∞ Ad (exp(tkXN))`k|b converges to `|b. In particular −λktk = Ad (exp(tkXN))`k(Xj+1) → 0 ` (Xj+1). Hence the sequence (tk)k converges to some t ∈ R and π`0 = π`. Similarly, we see 1 from (3.17) that for f ∈ Lc (GN ), the mapping ` → π`(f) is norm continuous when restricted j to the sets SN , j = 1, ··· ,N −2, since for the sequence (π`k )k above, the functions f`k converge in the L2-norm to f`.

2 Deﬁnition 3.8. Deﬁne for t, s ∈ R the selfadjoint projection operator on L (R) given by

2 Mt,sξ(x) := 1(t−s,t+s)(x)ξ(x), x ∈ R, ξ ∈ L (R), where 1(a,b), a, b ∈ R, denotes the characteristic function of the interval (a, b) ⊂ R.

20 We put for s ∈ R Ms := M0,s.

More generally, for a measurable subset T ⊂ R, we let MT be the multiplication operator with the characteristic function of the set T . For r ∈ R, let U(r) be the unitary operator on 2 L (R) deﬁned by

2 U(r)ξ(s) := ξ(s + r), ξ ∈ L (R), s ∈ R.

Deﬁnition 3.9. Let (π`k )k be a properly converging sequence in GbN with perfect data k k k n 2 ((ti )k, (ρi ), (si )). Let i ∈ C and let η ∈ D(R ) such that η has L -norm 1. Deﬁne for k 2 ρi , k ∈ N, i ∈ C, and u = (a, b) ∈ R the Schwartz function i s 2πia·s η(i, k, u)(s) := η(skp k + skb)e , s ∈ R. ρi By Example 3.2, for N = 3 we have

2πia·s η(1, k, u) = η(±sk|λk|s + sk(1 + b))e .

Let Pi,k,u be the operator of rank one deﬁned by

2 Pi,k,uξ := hξ, η(i, k, u)iη(i, k, u), ξ ∈ L (R).

2 Deﬁnition 3.10. For an element ϕ ∈ S(R ) let R ν(ϕ)(i, k) := sk 2 ϕˆ(a, −b)Pi,k,udadb, k ∈ , i ∈ C. R N 2 2 Then for ϕ ∈ S(R ), ξ ∈ L (R), s ∈ R, we have that Z ν(ϕ)(i, k)(ξ)(s) := sk ϕˆ(a, −b)(Pi,k,uξ)(s)du 2 R Z Z ! i t −2πia·(t−s) = sk ϕˆ(a, −b) ξ(t)η(skp + skb)e dt 2 k R R ρi i s η(skp k + skb)dbda ρi Z Z 2 i t i s = sk ϕˆ (s − t, −b)ξ(t)η(skp k + skb)η(skp k + skb)dtdb R R ρi ρi Z Z b t = ϕˆ2(s − t, − + pi )η(b)(3.18) s k R R k ρi i s i t η(sk p k − p k + b)ξ(t)dtdb. ρi ρi

21 Since η has L2-norm 1, using (3.17) and (3.18) we get k k (U(ti ) ◦ π`k (f) ◦ U(−ti ) ◦ Msk − ν(F (f)(0))(i, k) ◦ Msk )(ξ)(s) Z sk Z b t = ( fˆ2(s − t, (t + tk) · ` )) − fˆ2(s − t, − + pi , 0 ...) i k s k −sk R k ρi i s i t (3.19) η(b)η(sk p ( k ) − p k + b)db)ξ(t)dt ρi ρi Z sk Z ˆ2 k + ( f (s − t, (t + ti ) · `k))η(b) −sk R i s i t (η(b) − η(sk p k − p k + b))db)ξ(t)dt. ρi ρi ∗ 2 Proposition 3.11. Let ϕ ∈ C (R ), i ∈ C and k ∈ N. Then

1. the operator ν(ϕ)(i, k) is compact and kν(ϕ)(i, k)k ≤ kϕk ∗ 2 ; op C (R ) 2. we have that ν(ϕ)(i, k)∗ = ν(ϕ∗)(i, k); 3. furthermore

lim kν(ϕ)(i, k) ◦ (I − Ms ρk )kop = 0 k→∞ k i and hence

lim k(I − Ms ρk ) ◦ ν(ϕ)(i, k) ◦ Ms ρk kop = 0. k→∞ k i k i 2 Proof. 1.) It suﬃces to prove this for ϕ ∈ D(R ). We have that

Z Z Z b t s kν(ϕ)(i, k)ξk2 = | ϕˆ2(s − t, − )ξ(t)η(s pi( ) + b)dtη(s pi( ) + b)db|2ds 2 s k k k k R R R k ρi ρi Z Z b s = | (ϕ ˆ2(−, − ) ∗ (ξη ))(s)η(s pi( ) + b)db|2ds s k,b k k R R k ρi i t (where ηk,b(t) := η(skp ( k ) + b), t ∈ R) ρi Z 2 b 2 ≤ |(ϕ ˆ (−, − ) ∗ (ξηk,b))(s)| dbds 2 sk R Z 2 2 ≤ kϕk ∗ 2 kξηk,bk db C (R ) 2 ZR 2 2 i t 2 = kϕkC∗( 2) |ξ(t)| |η(skp ( ) + b)| dbdt R 2 k R ρi 2 2 = kϕk ∗ 2 kξk . C (R ) 2 Furthermore, since ν(ϕ)(i, k) is an integral of rank one operators, ν(ϕ)(i, k) must be compact. ∗ 2 Hence for every ϕ ∈ C ( ), ν(ϕ)(i, k) is a compact operator bounded by kϕk ∗ 2 . R C (R ) 2 2.) Let ϕ ∈ S(R ). Then ϕˆ = ϕˆ∗ and so Z Z ∗ ∗ ν(ϕ)(i, k) = (sk ϕˆ(u)Pi,k,udu) = sk ϕˆ(u)Pi,k,udu 2 2 Z R R ∗ ∗ = sk ϕˆ (u)Pi,k,udu = ν(ϕ )(i, k). 2 R

22 2 k k c 3.) Take now ϕ ∈ S(R ), such thatϕ ˆ has a compact support. We denote by [−skρi , skρi ] k k 2 the set R \ [−skρi , skρi ]. By (3.18) for any ξ ∈ L (R), s ∈ R we have

ν(ϕ)(i, k) ◦ ( − M k )(ξ)(s) I skρi Z Z 2 b i t i s i t = ϕˆ (s − t, − + p ( k ))η(b)η(sk(p ( k ) − p ( k )) + b)dbξ(t)dt = 0 k k c sk ρ ρ ρ [−skρi ,skρi ] R i i i 2 b i t k k c since for k large enoughϕ ˆ (s − t, − s + p ( k )) = 0 for any t ∈ [−skρi , skρi ] , b ∈ supp(η), k ρi s ∈ . Hence ν(ϕ(i, k))◦( −M k ) = 0 for k large enough. Since the mapping ν is continuous, R I skρi ∗ 2 it follows that limk→∞ kν(ϕ)(i, k)( − M k )kop = 0 for all ϕ ∈ C ( ) and every i ∈ C. I skρi R Hence also

lim k(I − Ms ρk ) ◦ ν(ϕ)(i, k) ◦ Ms ρk kop k→∞ k i k i ∗ = lim k(Ms ρk ◦ ν(ϕ )(i, k) ◦ (I − Ms ρk )kop k→∞ k i k i ∗ ≤ lim kν(ϕ )(i, k) ◦ (I − Ms ρk )kop = 0. k→∞ k i

2 gen ∗ 2 Definition 3.12. Let Let A = (A(`) ∈ K(L (R)), ` ∈ SN ,A(0) ∈ C (R )) be a field of bounded operators. We say that A satisfies the generic condition if for every properly ˆ converging sequence with perfect data (π`k )k ⊂ GN and for every limit point πì , i ∈ D, and for every adapted real sequence (sk)k k k i (3.20) lim kU(ti ) ◦ A(`k) ◦ U(−ti ) ◦ Msk − A(` ) ◦ Msk kop = 0. k→∞ A satisfies the character condition if for every properly converging sequence with perfect data gen (π`k )k, `k ∈ SN and for every limit point πì , i ∈ C, and for every adapted real sequence (sk)k k k lim kU(ti ) ◦ A(`k) ◦ U(−ti ) ◦ Ms ρk − ν(A(0))(i, k) ◦ Ms ρk kop = 0. k→∞ k i k i gen A satisfies the infinity condition, if for any properly converging sequence (π`k ), `k ∈ SN , with perfect data we have that

lim kA(`k) ◦ MT kop = 0, k→∞ k Sm k k k k gen where Tk = R \ i=1[ti − skρi , ti + skρi ] , and that for every sequence (`k)k ⊂ SN , for which the sequence of orbits GN · `k goes to inﬁnity we also have

lim A(`k) = 0. k→∞ ∗ ∗ We can now define the operator field C -algebra DN , which will be the image of the Fourier ∗ transform of C (GN ). ∗ Definition 3.13. Let DN be the space of all bounded operator fields A = (A(`)) ∈ 2 gen ∗ 2 ∗ K(L (R)), ` ∈ SN ,A(0) ∈ C (R ), such that A and the adjoint field A satisfy the generic, ∗ the character and the infinity conditions. Let for A ∈ DN gen kAk := sup{kA(`)k , kA(0)k ∗ 2 : ` ∈ S }. ∞ op C (R ) N

23 ∗ It is clear that DN is a Banach space for the norm k · k∞, since the generic, the character and the inﬁnity conditions are stable for the sum, for scalar multiplication and limits of sequences of operator ﬁelds.

∗ Theorem 3.14. Let a ∈ C (GN ) and let A be the operator field defined by A = F (a) as in Definition 3.6. Then A satisfies the generic, the character and the infinity conditions.

1 Proof. For the infinity condition, it suffices to remark that for any f ∈ Lc (GN ), and k large ˆ2 enough, we have that f (s − t, t · `k) = 0 for every s ∈ R, t ∈ Tk and so π`k (f) ◦ MTk = 0. If GN · `k goes to infinity in the orbit space, then R · `k is outside any given compact subset ∗ ˆ2 K ⊂ gN and so f (s − t, t · `k) = 0, s, t ∈ R and hence π`k (f) = 0 for k large enough. Using the density argument, we see that the infinity condition is satisfied for every element in the ∗ Fourier transform of C (GN ). For the generic condition, let (`k)k be a properly converging sequence in SN with perfect data. 1 2 Take i ∈ D. Then for an adapted sequence (sk)k, f ∈ Lc (GN ) and ξ ∈ L (R), s ∈ R, we have that

k k (U(ti ) ◦ π`k (f) ◦ U(−ti ) ◦ Msk − π`i (f) ◦ Msk )ξ(s) Z sk ˆ2 k ˆ2 i (3.21) = f (s − t, (t + ti ) · `k)) − f (s − t, t · ` )) ξ(t)dt. −sk

i ˆ Let pk and qi be the polynomials corresponding to `k and ` respectively, i.e., pk(t) = `k(t) i sk and qi(t) = `ˆ (t). Since lim → 0, j ∈ J(i), there exists R > 0 such that (s − t, (t + k→∞ k k |ti − aj | k k 0 k ˆ2 ti ) · `k) = (s − t, pk(t + ti ), −pk(t + ti ),...) is out of the support of f if t ∈ [−sk, sk] and |t| > R. In fact if t ∈ [−sk, sk] we have

d Y Y Y t Y |p (t + tk)| = |c (t + tk − ak)| = |c |tk − ak| | + 1| |t + tk − ak| k i k i j k i j tk − ak i j j=1 j∈J(i) j∈J(i) i j j∈ /J(i) Y sk Y k k ≥ |bi| |1 − | |t + t − a |, |tk − ak| i j j∈J(i) i j j∈ /J(i) where bi is the leading coeﬃcient of the polynomial qi, giving the statement. Thus by (3.21)

k k (U(ti ) ◦ π`k (f) ◦ U(−ti ) ◦ Msk − π`i (f) ◦ Msk )ξ(s) Z R ˆ2 k ˆ2 i = f (s − t, (t + ti ) · `k)) − f (s − t, t · ` )) ξ(t)dt −R

k k for k large enough. It is clear now that U(ti ) ◦ π`k (f) ◦ U(−ti ) ◦ Msk − π`i (f) ◦ Msk converges to 0 with respect to the Hilbert-Schmidt norm and hence in the operator norm. ∗ 1 Let a ∈ C (GN ). Then for any ε > 0 there exists f ∈ Lc (GN ) such that ||π(f) − π(a)||op ≤ ∗ gen ∗ ||f − a||C (GN ) < ε for any representation π of C (GN ). Thus for A(`) = π`(a), ` ∈ SN we have

k k i k k kU(ti ) ◦ A(`k) ◦ U(−ti ) ◦ Msk − A(` ) ◦ Msk kop = kU(ti ) ◦ (A(`k) − πlk (f)) ◦ U(−ti )kop k k i +kU(ti ) ◦ π`k (f) ◦ U(−ti ) ◦ Msk − πì (f) ◦ Msk kop + k(πì (f) − A(` ))kop → 0, and hence A satisfies the generic condition.

24 2 1 Choose now i ∈ C. By (3.19), for k ∈ N, s ∈ R, ξ ∈ L (R), f ∈ Lc (GN )

k k U(t ) ◦ π` (f) ◦ U(−t ) ◦ M k − ν(F (f(0)))(i, k) ◦ M k (ξ)(s) i k i skρi skρi k Z skρi Z ˆ2 k ˆ2 b i t = ( f (s − t, (t + ti ) · `k)) − f (s − t, − + p ( k ), 0 ..., 0) k sk ρ −skρi R i i s i t η(b)η(sk(p ( k ) − p ( k )) + b)db)ξ(t)dt + ρi ρi Z sk Z ˆ2 k i s i t ( f (s − t, (t + ti ) · `k))η(b)(η(b) − η(sk(p ( k ) − p ( k )) + b))db)ξ(t)dt. −sk R ρi ρi In order to show that

k k (3.22) kU(t ) ◦ π` (f) ◦ U(−t ) ◦ M k − ν(F (f(0)))(i, k) ◦ M k k → 0, k → ∞, i k i skρi skρi consider

k Z skρi ˆ2 k ˆ2 b i t q(k, i)(s, b) = (f (s − t, (t + ti ) · `k)) − f (s − t, − + p ( k ), 0 ..., 0)) k s ρ −skρi k i i s i t η(sk(p ( k ) − p ( k )) + b))ξ(t)dt = u(k, i) + v(k, i), ρi ρi where

k Z skρi ˆ2 k 0 k ˆ2 k u(k, i)(s, b) = (f (s − t, pk(t + ti ), −pk(t + ti ),...) − f (s − t, pk(t + ti ), 0,...)) k −skρi i s i t η(sk(p ( k ) − p ( k )) + b))ξ(t)dt, ρi ρi k Z skρi ˆ2 k ˆ2 b i t v(k, i)(s, b) = (f (s − t, pk(t + ti ), 0,...) − f (s − t, − + p ( k ), 0 ...)) k s ρ −skρi k i i s i t η(sk(p ( k ) − p ( k )) + b))ξ(t)dt. ρi ρi and let

Z Z sk Z ˆ2 k i s i t w(k, i)(s) = ( f (s − t, (t + ti ) · `k))η(b)(η(b) − η(sk(p ( k ) − p ( k )) + b))db)ξ(t)dt. R −sk R ρi ρi Our aim is to prove that for p(s, b) = 1 (s, b) R×supp(η)

(3.23) ku(k, i)pk2 ≤ ωk||ξ||2, kv(k, i)pk2 ≤ δk||ξ||2 and kw(k, i)k2 ≤ rk||ξ||2

with ωk, δk, rk → 0 as k → ∞. This will imply Z Z 2 2 2 2 | (q(k, i)(s, b))η(b)db| ds ≤ kq(k, i)k2kηk2 ≤ (ωk + δk) kξk2 R R which together with kw(k, i)k2 ≤ rk||ξ||2 will give (3.22).

25 k skρi k To see this we note ﬁrst that since j k → 0 if j∈ / L(i), we have that for |t| ≤ skρi |ak−ti |

d Y Y Y t Y t |p (t + tk)| = |c (t + tk − ak)| = |c |tk − ak| | + 1| | + 1)| k i k i j k i j tk − ak tk − ak j=1 j j∈ /L(i) i j j∈L(i) i j k k Y skρ Y |t| ρ ≥ σ |1 − i | | i − 1| |tk − ak| ρk |tk − ak| j∈ /L(i) i j j∈L(i) i i j

ˆ2 k 0 for some σ > 0. Thus for large k there exists R > 0 such that f (s − t, pk(t + ti ), −pk(t + k ˆ2 k k k ti ),...) = 0 and f (s − t, pk(t + ti ), 0,...) = 0 if |t| < skρi and |t| > Rρi . Hence the k k integration over the interval [−sk, sk] can be replaced by the integration over [−Rρi , Rρi ] in 1 the expression for u(k, i), v(k, i)p and w(k, i). Since f ∈ Lc (GN ) we have that

for some constant C > 0 and m ∈ N, m ≥ 2 . This gives

Rρk Z Z i s t 2 i i kv(k, i)pk2 = C η(sk(p ( k ) − p ( k ) + b) 2 k ρ ρ R −Rρi i i 2 k i t b ξ(t) (pk(t + ti ) − p ( k ) + ) m dt dsdb ρi sk 1 + |t − s| k Z Z Rρi 3C i s i t ≤ 2 η(sk(p ( k ) − p ( k ) + b) s 2 k ρ ρ k R −Rρi i i 2 i s i t ξ(t) (b + sk(p ( k ) − p ( k )) m dt dsdb ρi ρi 1 + |t − s| k 2 Z Z Rρi i s i t k i t ξ(t) +3C η(sk(p ( k ) − p ( k )) + b)(pk(t + ti ) − p ( k )) m dt dsdb 2 k ρ ρ ρ 1 + |t − s| R −Rρi i i i k 2 Z Z Rρi i s i t i t i s 1 +3C ξ(t)η(sk(p ( k ) − p ( k )) + b)(p ( k ) − p ( k )) m dt dsdb 2 k ρ ρ ρ ρ 1 + |t − s| R −Rρi i i i i k 2 Z Z Rρi C1 i s i t 1 ≤ 2 ξ(t)˜η(sk(p ( k ) − p ( k )) + b) m dtdsdb s 2 k ρ ρ 1 + |t − s| k R −Rρi i i (whereη ˜(b) = bη(b)) k 2 Z Z Rρi i s i t k i t +C1 ξ(t)η(sk(p ( k ) − p ( k )) + b)(pk(t + ti ) − p ( k )) 2 k ρ ρ ρ R −Rρi i i i 1 dtdsdb 1 + |t − s|m k 2 Z Z Rρi i s i t i t i s + ξ(t)η(sk(p ( k ) − p ( k )) + b)(p ( k ) − p ( k )) 2 k ρ ρ ρ ρ R −Rρi i i i i 1 dtdsdb 1 + |t − s|m

k k i As pk(tρi + ti ) − p (t) converges to 0 uniformly on each compact, Z R k 2 k k i 2 k 2 |ξ(tρi )| |pk(tρi + ti ) − p (t)| ρi dt ≤ rkkξk2 −R t s t − s X t t − s with r → 0 as k → ∞. Moreover, pi( ) − pi( ) = α ( )β ( ) for some ﬁnite k ρk ρk ρk l ρk l ρk i i i l i i number of polynomials αl, βl which do not depend on k. Thus

k 2 Z Z Rρi 2 i t i s 1 |ξ(t)| p ( k ) − p ( k ) m dtds k ρ ρ 1 + |t − s| R −Rρi i i 2 Rρk Z Z i t − s t t − s 1 2 X = |ξ(t)| k αl( k )βl( k ) m dtds k ρ ρ ρ 1 + |t − s| R −Rρi i l i i k Z Rρi C4 2 C4 2 ≤ k |ξ(t)| dt ≤ k kξk2 (ρ )2 k (ρ )2 i −Rρi i for a properly chosen m. It follows now that

||v(k, i)p||2 ≤ δkkξk2 for some δk → 0 as k → ∞. For w(k, i) we have

Rρk Z Z i Z 2 ˆ2 k kw(k, i)k = f (s − t, (t + ti ) · `k)η(b) k R −Rρi R 2 i s i t (η(b) − η(sk(p ( k ) − p ( k )) + b))ξ(t)dbdt ds ρi ρi 2 Rρk Z Z i s t 1 2 i i ≤ Ckηk2 |sk(p ( k ) − p ( k ))| m |ξ(t)|dt ds k ρ ρ 1 + |t − s| R −Rρi i i k Z Z Rρi 2 i s i t 2 1 2 ≤ Ckηk2 |sk(p ( k ) − p ( k ))| m |ξ(t)| dtds k ρ ρ 1 + |t − s| R −Rρi i i for some constant C. Then using the previous arguments we get

Ds2 kw(k, i)k2 ≤ k kξk2kηk2. k 2 2 2 (ρi )

sk As k → 0 we get the desired inequality for w(k, i). ρi

27 To prove the inequality for u(k, i) we have as in the previous case that

ˆ2 k 0 k ˆ2 k |f (s − t, pk(t + ti ), −pk(t + ti ),...) − f (s − t, pk(t + ti ), 0,...)| N−2 X (n) 1 ≤ C( |p (t + tk)|2)1/2 k i 1 + |t − s|m n=1

(n) for some constant C > 0 and m ∈ N, m ≥ 2; here pk denotes the n-th derivative of pk. For n = 1 we have

d−1 Y X 1 Y t 1 |t| |p0 (t + tk)| = |c (tk − aj )| ( + 1)| ≤ σ + 1 k i k i k (tk − al ) k j ρk ρk j l i k j6=l ti − ak i i

(n) k for some constant σ > 0. Similar inequalities hold for higher order derivatives pk (t + ti ) which show that

k k lim kU(ti ) ◦ π`k (f) ◦ U(−ti ) ◦ Ms ρk − ν(F (f(0)))(i, k) ◦ Ms ρk kop = 0. k→∞ k i k i

∗ To show now that the character condition holds for the ﬁelds A ∈ C\(GN ) we use again the 1 ∗ density of Lc (GN ) in C (GN ).

Corollary 3.15. Let (π`k )k be a properly converging sequence in GbN with perfect data k k k ∗ 2 ((ti )k, (ρi ), (si )). Let i ∈ C. Then for every ϕ, ψ ∈ C (R ) we have that

lim kν(ϕ)(i, k) ◦ ν(ψ)(i, k) − ν(ϕψ)(i, k)kop = 0. k→∞

2 Proof. Indeed, if we take ﬁrst ϕ, ψ in S(R ), then we can choose f, g ∈ S(GN ), such that

28 ρ(f) = ϕ, ρ(g) = ψ and so, by Proposition 3.11 and Theorem 3.14,

kν(i, k)(ϕ) ◦ ν(i, k)(ψ) − ν(i, k)(ϕψ)kop

≤ k(ν(i, k)(ϕ) ◦ ν(i, k)(ψ) − ν(i, k)(ϕψ)) ◦ M k kop skρi + k(ν(i, k)(ϕ) ◦ ν(i, k)(ψ) − ν(i, k)(ϕψ)) ◦ ( − M k )kop I skρi ≤ k(ν(i, k)(ϕ) ◦ ν(i, k)(ψ) k k k k −(U(t ) ◦ π` (f) ◦ U(−t )) ◦ (U(t ) ◦ π` (g) ◦ U(−t ))) ◦ M k kop i k i i k i skρi k k + k(U(t ) ◦ π` (f ∗ g) ◦ U(−t ) − ν(i, k)(ϕψ)) ◦ M k kop i k i skρi + k(ν(i, k)(ϕ) ◦ ν(i, k)(ψ) − ν(i, k)(ϕψ)) ◦ ( − M k )kop I skρi k k ≤ k(ν(i, k)(ϕ) − (U(t ) ◦ π` (f) ◦ U(−t )) ◦ ( − M k ) ◦ ν(i, k)(ψ) ◦ M k kop i k i I skρi skρi k k + k(ν(i, k)(ϕ) − (U(t ) ◦ π` (f) ◦ U(−t )) ◦ M k ◦ ν(i, k)(ψ) ◦ M k kop i k i skρi skρi k k k k + k((U(t ) ◦ π` (f) ◦ U(−t )) ◦ (ν(i, k)(ψ) − (U(t ) ◦ π` (g) ◦ U(−t )) ◦ M k kop i k i i k i skρi k k + k(U(t ) ◦ π` (f ∗ g) ◦ U(−t ) − ν(i, k)(ϕψ)) ◦ M k kop i k i skρi + k(ν(i, k)(ϕ) ◦ ν(i, k)(ψ) − ν(i, k)(ϕψ)) ◦ ( − M k )kop → 0 as k → ∞. I skρi

∗ 2 The usual density condition shows that the statement holds for all ϕ, ψ ∈ C (R ). ∗ ∗ ∗ Theorem 3.16. The space DN is a C -algebra, which is isomorphic with C (GN ) for every N ∈ N,N ≥ 3. ∗ ∗ ∗ Proof. Let us first show that DN is a C -algebra. We prove first that DN is closed under multiplication. Let A = (A(`), ` ∈ SN ) and B = (B(`), ` ∈ SN ) satisfy the generic condition ˆ and let (π`k )k ⊂ GN be a properly convergent sequence with perfect data such that for every limit point πli , i ∈ D, and for every adapted real sequence (sk)k the fields A, B satisfy (3.20). Then

k k i i kU(ti ) ◦ A(`k) ◦ B(`k) ◦ U(−ti ) ◦ Msk − A(` ) ◦ B(` ) ◦ Msk kop k k i k k ≤ k(U(ti ) ◦ A(`k) ◦ U(−ti ) ◦ Msk − A(` ) ◦ Msk ) ◦ U(ti ) ◦ B(`k) ◦ U(−ti ) ◦ Msk kop i k k i +kA(` ) ◦ Msk ◦ (U(ti ) ◦ B(`k) ◦ U(−ti ) ◦ Msk − B(` ) ◦ Msk )kop k k k k i +kU(ti ) ◦ A(`k) ◦ U(−ti ) ◦ (I − Msk ) ◦ (U(ti ) ◦ B(`k) ◦ U(−ti ) ◦ Msk − B(` ) ◦ Msk )kop i i +kA(` ) ◦ (I − Msk ) ◦ B(` ) ◦ Msk kop k k i +kU(ti ) ◦ A(`k) ◦ U(−ti ) ◦ (I − Msk ) ◦ B(` ) ◦ Msk kop.

i i Since B(` ) is compact and I − Msk converges to 0 strongly, k(I − Msk ) ◦ B(` )kop → 0 giving that the product A(`) ◦ B(`) satisfies the generic condition. To see that the character condition is closed under multiplication we argue as before, but use k( − M k ) ◦ ν(ϕ)(i, k) ◦ M k kop → 0 which is due to Propsition 3.11. I skρi skρi The infinity condition is clearly closed under multiplication of fields. ∗ ∗ By Theorem 3.14, the Fourier transform F maps C (GN ) into DN Let us show that F is also onto. By the Stone-Weierstrass approximation theorem, we must only prove that the dual ∗ ∗ space of DN is the same as the dual space of C (GN ). We proceed by induction on N. If N = 3, then GN is the Heisenberg group and the statement follows from Theorem 2.16. Let ∗ π ∈ DdN .

29 ∗ ∗ Let for M = 3, ··· N − 1, RM : DN → DM be the restriction map, i.e. denote by qM : gN → t ∗ ⊥ ∗ gN /bN−M ' gM the quotient map and by qM : gM ' bN−M → gN its transpose. Then for an ∗ gen operator field A ∈ DN we define the operator field RM (A) over SM by: ˜ t ˜ ˜ gen RM (A)(`) := A(qM (`)), ` ∈ SM . ∗ ∗ It follows from the definition of DN that the image of RM is contained in DM . Hence RM is ∗ a homomorphism of C -algebras, whose kernel IM is the ideal ∗ ⊥ IM = {A ∈ DN ,A(`) = 0 for all ` ∈ SN ∩ bN−M }. ∗ ∗ ∗ Let QM : C (GN ) → C (GM ) ' C (GN /BN−M ) be the canonical projection. Then the kernel ∗ ⊥ of this projection is the ideal JM := {a ∈ C (GN ); π`(a) = 0, ` ∈ SN ∩ bN−M }. Let us write ∗ ∗ FM for the Fourier transform C (GM ) → DM . With these notations we have the formula ∗ (3.24) RM (FN (a)) = FM (a modulo JM ), a ∈ C (GN ).

∗ ∗ Since by the induction hypothesis DdM = FM (C\(GM )) for every 3 ≤ M ≤ N − 1 we see from ∗ ∗ ∗ (3.24) that RM (FN (C (GN ))) = FM (C (GM )) = DM and so the mapping RM is surjective ∗ ∗ for such an M. Hence DN /IM ' C (GM ). We have also IN−1 ⊆ IN−2 ⊆ ... ⊆ I3. If π(IN−1) = {0} then π ∈ G\N /B1 ⊂ GdN . 1 2 Suppose now that π(IN−1) 6= {0}. Let us show that IN−1 ' C0(SN , K(L (R))). It is clear from ∗ 1 2 ∗ the definition of DN that C0(SN , K(L (R))) ⊂ DN and so is contained in IN−1. It suffices to 1 2 show now that IN−1 ⊂ C0(SN , K(L (R))). For that it is enough to see that for any element 1 A in IN−1 and any sequence (`k)k in SN for which either (π`k ) converges to infinity or to a 1 representation π` with ` 6∈ SN , we have that limk kA(`k)kop = 0. This follows from the infinity 1 condition in the first case. In the second case no limit point of the sequence (π`k ) is in SN by Remark 3.7. It suffices to show then that limk kA(`k)kop = 0 for every subsequence with perfect data (also indexed by k for simplicity of notation). We have with the notations of Definition 3.12 that for k ∈ N

A(`k) = A(`k) ◦ MSk + A(`k) ◦ MTk . k k k k where Sk = ∪i(ti − skρi , ti + skρi ), Tk = R \ Sk.

Since A(`) = 0 for every π` in the limit set of the sequence (π`k )k, the generic and the character conditions say that k k lim kU(ti ) ◦ A(`k) ◦ U(ti ) ◦ Ms ρk kop = 0. k k i Hence

lim kA(`k) ◦ Mtk,s ρk kop = 0 k i k i and since also lim kA(`k) ◦ MT kop = 0 k k 1 2 it follows that limk kA(`k)kop = 0. Hence IN−1 ⊂ C0(SN , K(L (R))) and so IN−1 = 1 2 1 C0(SN , K(L (R))). Finally π|IN−1 is evaluation in some point ` ∈ SN and so π ∈ GdN . This ﬁnishes the proof of the theorem.

Acknowledgements. We would like to thank K. Juschenko for the reference [Gor]. The second author was supported by the Swedish Research Council.

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Jean Ludwig, Laboratoire LMAM, UMR 7122, Département de Mathématiques, Université Paul Verlaine Metz, Ile de Saulcy, F-57045 Metz cedex 1, France, [email protected]

Lyudmila Turowska, Department of Mathematics, Chalmers University of Technology and University of Gothenburg, SE-412 96 G¨oteborg, Sweden, [email protected]