The C*-algebra of the Boidol group

The Boidol group is the smallest non-*-regular exponential Lie group. It is of dimension 4 and its Lie algebra is an extension of the Heisenberg Lie algebra by the reals with the roots 1 and -1. We describe the C*-algebra of the Boidol group as an algebra of operator fields defined over the spectrum of the group. It is the only connected solvable Lie group of dimension less than or equal to 4 whose group C*-algebra had not yet been determined.


Introduction and notations
Let A be a C*-algebra and A be its spectrum.In order to analyse the C*-algebra A, one can use the Fourier transform F , which will allow us to decompose A over A. To be able to define such a transform, we choose a representative π in every equivalence class [π] of A and we consider the C*-algebra l ∞ ( A) of all bounded operator fields defined over A given by where H π is the Hilbert space of π.The Fourier transform F of A is defined by It is then an injective, hence isometric, homomorphism from A into l ∞ ( A). Therefore we can analyse the C*-algebra A by recognising the elements of F (A) inside the (big) C*-algebra l ∞ ( A).However, for most C*-algebras its spectrum is not known or the topology of its spectrum is a mystery.
In the case of the C*-algebra C * (G) of a locally compact group G, we know that the spectrum C * (G) of C * (G) can be identified with the unitary dual G of G. Furthermore, if G is an exponential Lie group, i.e., it is a connected, simply connected solvable Lie group for which the exponential mapping exp : g → G from the Lie algebra g to its Lie group G is a diffeomorphism, then the Kirillov-Bernat-Vergne-Pukanszky-Ludwig-Leptin theory shows that there is a canonical homeomorphism K : g * /G → G from the space of coadjoint orbits of G in the linear dual space g * onto the unitary dual space G of G (see [11] for details and references).In this case, one can identify the spectrum C * (G) of the C*-algebra of an exponential Lie group G with the space g * /G of coadjoint orbits of the group G.Note that connected Lie groups are second countable, so the algebra C * (G) and its dual space G are separable topological spaces.
The idea of describing group C*-algebras as algebras of operator fields defined on the dual spaces was first introduced in [5] and [9].In serial, the C*-algebra of ax + b-like groups [10], of the Heisenberg groups and of the threadlike groups [12], of the affine automorphism groups G n,µ in [6], [7], and of the group T ⋉ H 1 [13] have all been characterised as algebras of operator fields defined on the corresponding spectrums of the groups.Note that each case has its own treatment due to the complexity of the coadjoint orbits of each group.In this way, the C*-algebra of every exponential Lie group of dimension less than or equal to 4 has been explicitly determined with one exception, namely the Boidol group, which is an extension of the Heisenberg group by the reals with the roots 1 and −1.
In this paper, we consider this Boidol group G = exp g, which is the only non- * -regular exponential Lie group of dimension 4 (see [3]).We will write down precisely the dual space G of Boidol's group G = exp g using the structure of the coadjoint orbits in g * /G and determine its topology.We decompose this orbit space into the union of a finite sequence (Γ i = S i \ S i−1 ) d i=0 , where d = 3, of relatively closed subsets.On each of the sets Γ i , the orbit space topology is Hausdorff and the main question is to understand the operator fields â, a ∈ C * (G), in particular the behaviour of the operators â(γ), γ ∈ Γ i , when γ approaches elements in Γ i−1 .For each of these sets Γ i , we obtain different conditions for the group C*-algebra.Since the spectrum of Boidol's group has more layers than the spectra of the other groups of dimension less than or equal to 4, the analysis of the behaviour of the operator fields â, a ∈ C * (G), is more involved.
We first recall the following definitions which were given in [1].Let H be a Hilbert space and B(H) denote the algebra of bounded linear operators on H. Definition 1.1.Let d be a natural number.
(1) Let S be a topological space.We say that S is locally compact of step ≤ d, if there exists a finite increasing family ∅ = S 0 ⊂ S 1 ⊂ • • • ⊂ S d = S of closed subsets of S, such that the subsets Γ 0 = S 0 and Γ i := S i \ S i−1 , i = 1, . . ., d, are locally compact and Hausdorff in their relative topologies.(2) Let S be locally compact of step ≤ d and {H i } i=0,...,d be Hilbert spaces.For a closed subset M ⊂ S, denote by CB(M, H i ) the unital C*-algebra of all uniformly bounded operator fields (ψ(γ) ∈ B(H i )) γ∈M∩Γi,i=0,...,d , which are operator norm continuous on the subsets Γ i ∩ M for every i ∈ {0, . . ., d} with Γ i ∩ M = ∅, such that γ → ψ(γ) goes to 0 in operator norm if γ goes to infinity on M .We equip the algebra CB(M, H i ) with the infinity-norm (3) For every s ∈ S, choose a Hilbert space H s .We define the C*-algebra l ∞ (S) of uniformly bounded operator fields defined over S by Definition 1.2.Let A be a separable liminary C*-algebra such that the spectrum A of A is a locally compact space of step ≤ d, with closed subsets S i of A and Assume that for every i ∈ {0, . . ., d}, there exists a Hilbert space H i and a concrete realisation (π γ , H i ) of γ on the Hilbert space H i for every γ ∈ Γ i := S i \ S i−1 .Denote by F : A → l ∞ ( A) the Fourier transform of A into the C*-algebra l ∞ ( A) defined as Definition 1.1(3), i.e., for a ∈ A, let We say that F (A) is continuous of step ≤ d, if the set S 0 is the collection of all characters of A, and for every γ ∈ Γ i there is a concrete realisation (π γ , H i ) of γ on the Hilbert space H i such that

The Boidol group
2.1.Let g be the real Lie algebra of dimension 4 with a basis {T, X, Y, Z} and the non-trivial brackets The simply connected connected group G with Lie algebra g, which we call the Boidol group, can be realised on R 4 with the multiplication (t, x, y, z) The inverse of (t, x, y, z) is given by (t, x, y, z) −1 = (−t, −e −t x, −e t y, −z).
We see that the subgroup Z := {(0, 0, 0, z); z ∈ R} is the centre of G. Furthermore, is the semi-direct product of R acting on the Heisenberg group H = {0} × R 3 .

2.2.
The coadjoint orbit space.In this section we give a system of representatives of the coadjoint orbits in the linear dual space g * of g.Let {T * , X * , Y * , Z * } be the dual basis of {T, X, Y, Z}.We have three different kinds of coadjoint orbits in g * .
(1) The orbits in general position: The orbits of dimension 2 vanishing on Z: (3) The real characters: For every τ ∈ R we have the real character ℓ τ := τ T * of g.This gives us the following partition of g * , The subset Σ 2 of Γ 2 , defined by where ℓ τ := (τ, 0, 0, 0).We recall that (1) The spectrum of G/Z (and of C * (G/Z)) can be identified with the closed subset {π ∈ G; π(Z) = {I Hπ }} and also with the subset (2) Let (λ G/Z , L 2 (G/Z)) be the left regular representation of the group G/Z and ( λG/Z , L 2 (G/Z)) be the corresponding representation of G. Since G is amenable, the representation λ G/Z is injective in C * (G/Z).It is easy to see (either directly or using [14]) that λG/Z (C * (G)) = λ G/Z (C * (G/Z)).Furthermore the kernel of λG/Z in C * (G) is the ideal The C*-algebra C * (G/Z) is thus isomorphic to the quotient of C * (G) by K S2 .
(3) We observe that the group G/Z is an extension of R 2 by R with the roots +1 and −1, and its C*-algebra has been described in [10].(4) It follows from the description of the coadjoint orbits that the orbits in Γ 3 , Γ 2 and Γ 0 are closed in g * , but the four orbits in Γ  [11], page 135).
In the following we recall the definition of properly converging sequences and their limit sets, we also give a proposition of properly converging sequences in our group.Definition 2.3.Let S be a topological space.Let x = (x k ) k be a net in S. We denote by L(x) the set of all limit points of the net x.A net x is called properly converging if x has limit points and if every cluster point of the net is a limit point, i.e., the set of limit points of any subnet is always the same, indeed, equals to L(x).
We know that every converging net in S admits a properly converging subnet, hence, we can work with properly converging nets in our space.
Note that throughout the paper the symbols σ, ε will denote elements of the set {+1, −1}.
(I).For ℓ ρ,λ ∈ Γ 3 , consider the Pukanszky polarisation p := span{T, Y, Z} at ℓ ρ,λ .This gives us the irreducible representation π ρ,λ := ind G P χ ρ,λ , where P = exp p and χ ρ,λ (exp(tT ) exp(yY ) exp(zZ)) := e −i(ρt+λz) for t, y, z ∈ R is the unitary character of P corresponding to the character ℓ ρ,λ of p.The Hilbert space L 2 (G/P, χ ρ,λ ) of π ρ,λ can be realised as Therefore, π ρ,λ is a kernel operator with kernel function Furthermore, for u ∈ R, we obtain the identity e −iρt e −iλz e −iλxy/2 e −iλe t yu ξ(e t u + x), which shows that R) be the unitary operator defined by We shall need later in Section 4 an equivalent version of the representations π ρ,λ .Definition 3.2.For any (ρ, λ) ∈ R × R * and measurable function ϕ : R → R, the operator is a unitary operator.We have that We can also view V as an operator V : where Let us now take the measurable function ϕ as Hence, writing λ = ε|λ|, we obtain: where Using partial integration, we see that Hence, there exist positive continuous functions ψ, ψ ′ with compact support on R such that Furthermore, with the automorphism α on the group G and s ∈ R, we have that e iρ(ln(|−λx+εe t s|−ln(|e t s|)) |e t s| 1/2 | − λx + εe t s| 1/2 η(−ελx + e t s)dxdt.
(II).For ℓ ∈ Γ 2 Γ 1 , the subalgebra h := span{X, Y, Z} is a Pukanszky polarisation at ℓ. Therefore the unitary representation This means that π ℓ (F ) is a kernel operator with kernel function Let us write an equivalent representation for π ℓ : We use the multiplication invariant measure du Then U σ is a unitary operator and Let We obtain the relation: We see that On the other hand, In particular the representations τ σ µ,ν and τ −σ µ,ν are equivalent.Furthermore, for η ∈ L 2 (R σ , du |u| ) and F ∈ L 1 (G): x, y, z)e −iσµu −1 e −t x e −iσνue t y η(e t u)dtdxdydz In order to define the Fourier transform a → â, a ∈ C * (G), we identify G with the set Γ := Γ 3 ∪ Γ 2 ∪ Γ 1 ∪ Γ 0 and we let : 4. Norm control of dual limits.
In this section, we will describe the conditions that our C*-algebra as the image of the Fourier transform must fulfil, and characterise the C*-algebra of the Biodol group which will be our main result (Theorem 4.24).Let O = (O k ) k be a sequence in g * .We say that O tends to infinity, if for every bounded subset B ⊂ g * , the subsets O k ∩ Ad * (G)B of g * are empty for k large enough.This is equivalent to the property that no sequence (l k ∈ O k ) k∈N admits a convergent subsequence.
We have the following norm convergence condition.Proof.It suffices to consider only L 1 -functions F for which the partial Fourier transforms F 3,4 are contained in C ∞ c (R 4 ), since this space is dense in C * (G).The expression (2) tells us that the operators π ℓ (F ), ℓ ∈ Γ 3 , are Hilbert-Schmidt.Indeed for those functions F , the kernel functions F ρ,λ (u, x) are continuous in the parameters ρ, λ, u, x and have compact support in u, x ∈ R and |λ| > c (for some fixed positive real number c).Hence for any converging sequence (ℓ k ) k ⊂ Γ 3 with limit ℓ ∈ Γ 3 , the operators π ℓ k (F ) converge in the Hilbert-Schmidt norm to the operator π ℓ (F ).The cases in Γ 2 , Γ 1 and Γ 0 are treated in [10].
If now we have that lim k→∞ O k = ∞ in Γ 3 , then according to Proposition 2.4 we have that for some positive ψ ∈ C c (R) and any x, u ∈ R, hence lim k→∞ π ρ k ,λ k (F ) op = 0.In Γ 0 , Γ 1 and Γ 2 , the property of sequences of coadjoint orbits tending to infinity has been described in [10].

An approximation of π
Let (π ρ k ,λ k ) k be a properly converging sequence in G such that lim k→∞ λ k = 0 and that lim k→∞ λ k ρ k = ω ∈ R. We can assume (by passing to a subsequence if necessary) that λ k = ε|λ k | for every k ∈ N.
Then we have that Proof.By (1), it suffices to show that for Lemma 4.5.Suppose that lim k→∞ λ k = 0 and Then we have that Proof.We prove the lemma first for for s, x ∈ R and k ∈ N, where ψ is a positive continuous function with compact support on R given in (7).Now for any C > 0 large enough, we have that for another constant C ′′ > 0. Hence, Young's condition tells us that Now if we take any a ∈ C * (G), for every ε > 0 there exists an F ε in L 1 (G) with the properties above such that a − F ε C * (G) < ε, and then there exists Recall the equivalent representations τ ± µ,ν given in (13).We have the following lemma.
for some constant C depending on F .
Proof.Take any F ∈ L 1 (G) such that F 3,4 ∈ C c (R 4 ).Then F 3,4 (t, s, x, λ) = 0 for any t, s, x ∈ R such that |t| Hence for k large enough and for some new constant , we have: , for all our s, x, t in R. Whence .
By (6) we have that It follows for our constant C 1 that for any Therefore, for any η ∈ L 2 (R, dx |x| ) with η(x) = 0 for |x| ≤ R k |λ k | and k large enough, we deduce that Then we obtain the inequality: for some new constant C > 0 depending on F .Now e t s − ελ k x = e t sα k (t, s, x) = e t+ln(α k (t,s,x)) s, where η(e r s)dxµ ′ k (r)dr.
Now by ( 23), for k large enough, we have where ϕ ∈ C ∞ c (R, R + ) is given before (22).Therefore for k large enough and any η ∈ L 2 (R, ds |s| ) with η 2 ≤ 1 and η(u for a new constant C > 0 depending on F .Let us recall that for η ∈ L 2 (R σ , dx |x| ) (see ( 13)), Hence for any η ∈ L 2 (R + , ds |s| ), we have that for a constant C > 0. Then for any s ∈ R, it follows that Let (π ρ k ,λ k ) k be a properly converging sequence in G such that lim k→∞ λ k = 0 and lim k→∞ λ k ρ k = ω ∈ R * .We recall (Proposition 2.4) that the limit set L of the sequence (O ρ k ,λ k ) k is the two points set where τ σ µ,ν acting on L 2 (R σ , du |u| ) is given in (10).We can again assume that λ k = ε|λ k | for k ∈ N. By Proposition 2.4, the limit set L of the sequence of representations (π ρ k ,λ k ) k is equal to Γ 1 ∪ Γ 0 .We first show that a similar convergence as the one in Lemma 4.5 holds when ω = 0, but with a slightly different sequence Then we have that ).There exists M > 0 such that F 3,4 (t, x, y, λ) = 0 for k large enough.Choose an even function ϕ : R → R + in C c (R) with compact support such that Now, by Young's inequality, for some constant C ′ > 0.
Lemma 4.9.Let (ρ k ) k be a real sequences with lim k→∞ ρ k = 0. Then for any real sequence (λ k ) k we have that Proof.For k ∈ N, the identity where χ ρ k (exp tT • h) = e −iρ k t for t ∈ R and h ∈ H, the Heisenberg group, shows that where F 1 (t, x, y, z) := tF (t, x, y, z) and g = (t, x, y, z) ∈ G.
Lemma 4.10.Let (π ρ k ,λ k ) k be a properly converging sequence in G such that lim k→∞ λ k = 0 and for some constant C depending on F .
Proof.The statements (a) and (b) are proved in the same way as the corresponding ones in Lemma 4.6.
Remark 4.11.We can take, for example, In the second case we have that For the following arguments, we need to work with the multiplication operator M I , where I is a measurable subset of R, which acts on the space L 2 (R, dµ) with any Borel measure dµ on R. Definition 4.12.Suppose that lim k→∞ λ k = 0, ρ k = 0, k ∈ N, and k and I ± k,j , for k ∈ N and j ∈ {1, 2, 3}, in R by where the representation τ ± µ,ν is defined in (13).

This relation implies that lim
op = 0.In the same way, we have that η(e t )dt| We proceed similarly for the second relation.
4.5.The final theorem.Note that the representations in Γ 1 do not map C * (G) into the algebra of compact operators on L 2 (R + , dx |x| ).Following [10, Definition 6.2], we define the compact condition on l ∞ ( G), which will be satisfied by the elements of C * (G).
Remark 4.20.The operators σ 0,εω k ,ε k , s 0,ω k ,ε,± k and σ ω k are defined in Definition 4.14 and (25), respectively.Note that due to the limit set of a properly converging sequence may lie in different Σ j , j = 0, 1, 2, the operators involved in the approximations will depend on sequences (R k ) k in R + with different conditions given previously, and on sequences (Q k ) k , (P k ) k which give rise the internals I ± k,2 , J ± k used in Definition 4.14.

Proposition 4 . 2 .
For any a ∈ C * (G), the mappings O → π O (a) are norm continuous on the different sets Γ i for i = 0, 1, 2, 3.For any sequence (O k ) k tending to infinity, we have that lim k→∞ π O k (a) op = 0.

Remark 4 .
21. Since Z is the centre of Boidol's group G, we have seen in Remark 2.1 that the spectrum of G/Z (thus, of C * (G/Z)) can be identified with the subset S 2 := Γ 2 ∪ Γ 1 ∪ Γ 0 of the coadjoint orbit space, and the group G/Z is ax + b-like.It has been shown in[10, Section 8]  that the family of uniformly bounded operator fields defined over the set S 2 := Σ 2 ∪ Σ 1 ∪ Σ 0 satisfying the conditions (1), (3) and (4) in Definition 4.19 forms a C*-algebra ([10, Proposition 8.2]), denoted by D * (G/Z) for our purpose, and it is isomorphic to C * (G/Z) via the Fourier transform.Hence the natural restriction map R G/Z : C * (G) → C * (G/Z) is surjective, so is the natural quotient map P G/Z : C * (G) → C * (G/Z) (see Remark 2.1(2)).This shows that for every operator field φ ∈ D * (G/Z), there exists an a φ ∈ C * (G) such that φ coincides with a φ |S2 .That is, for every φ ∈ D * (G/Z) = F G/Z (C * (G/Z)), where F G/Z is the Fourier transform defined on the C*-algebra of G/Z (see also [10, Definition 5.3]), thanks to the isomorphism C * (G/Z) ≃ C * (G)/K S2 (given in Remark 2.1(2)) and the spectrum C * (G) can be identified with 3 i=0 Σ i which contains S 2 , we have such an a φ ∈ C * (G).Note that it follows from the preceding sections, mainly Proposition 4.2, Theorems 4.7, 4.15 and Remarks 4.18 and 4.21, that F (a) = â, a ∈ C * (G), satisfies all the conditions in D * (G).Thus, C * (G) ⊂ D * (G).

Condition ( 2
-d) follows in a similar way thanks to Theorem 4.15.Hence D * (G), being a closed * -subalgebra of l ∞ ( Ĝ), is a C*-algebra which contains the Fourier transform of C * (G).It is clear that the spectrum of D * (G) contains G since point evaluations are irreducible for C * (G).Take now π ∈ D * (G).Since there is a natural restriction map from C * (G) onto C * (G/Z) by Remark 4.21, and the Fourier transfer defined on C * (G) (resp.on C * (G/Z)) is an isometric homomorphism onto F G (C * (G)) which is contained in D * (G) (resp.F G/Z (C * (G/Z)) which coincides with D * (G/Z)), thus, there is a restriction map, denoted by R 0 , from D * (G) to D * (G/Z) which is also surjective.Let K 0 = {φ ∈ D * (G); φ(γ) = 0 for γ ∈ S 2 }.It can be seen easily that K 0 is an ideal of D * (G).Note that D * (G/Z) ≃ C * (G/Z) ≃ C * (G)/K S2 .It follows immediately from the conditions (1) and ( It is clear that D * (G) is postliminal, since every (non-trivial) irreducible representation (π, H π ) of D * (G) induces compact operators on H π (conditions (2-b), (3-b), (3-e)).Now we have our main theorem which characterises the C*-algebra C * (G) of Boidol's group by the above descriptions of the Fourier transform of C * (G) onto D * (G).

Theorem 4 .
24.The Fourier transform defined in (14) is an isomorphism of the C*-algebra of Boidol's group G onto the C*-algebra D * (G).Proof.Using Lemma 4.23, the Stone-Weierstrass theorem for C*-algebras (see[8, 11.1.8])tells us that the subalgebra C * (G) of D * (G) is equal to D * (G).
1are not.Since the canonical mapping K : g * /G → G is a homeomorphism, it follows that for every closed coadjoint orbit Ω ∈ g * /G the irreducible representations π Ω = K(Ω) associated to Ω send the C*-algebra of G onto the algebra of compact operators K(H πΩ ) on the Hilbert space H πΩ of π Ω .The relative topology on Γ 3 is Hausdorff.In Γ 2 the subsets Γ 2,ε,σ , ε, σ ∈ {+1, −1}, are open and Hausdorff, Γ 1 is discrete and Γ 0 is homeomorphic to R.Proof.The proof follows easily from the fact that a sequence of coadjoint orbits (O k ) k∈N ⊂ g * converges to an orbit O if and only if for every ℓ ∈ O and every k ∈ N there exists an ℓ k ∈ O k such that lim k→∞ ℓ k = ℓ (see R k and k large enough.Now if s, x have the same sign, we have that |εx + εe t s| ≥ R k and thus (16) is satisfied.Similarly, if s and x have different signs, it follows that