The C*-algebra of the Boidol group

Ying-Fen Lin; Jean Ludwig

doi:10.1515/forum-2021-0209

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The C*-algebra of the Boidol group

Ying-Fen Lin and Jean Ludwig

Published/Copyright: February 1, 2024

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From the journal Forum Mathematicum Volume 36 Issue 4

Abstract

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.

Keywords: C*-algebra; algebra of operator fields; spectrum

MSC 2020: 22D25; 47L10; 43A65

1 Introduction and notations

Let 𝒜 be a C*-algebra and let 𝒜 ^ be its spectrum. In order to analyze 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

l ∞ ( 𝒜 ^ ) := { A = ( A ( π ) ∈ ℬ ( ℋ π ) ) [ π ] ∈ 𝒜 ^ : ∥ A ∥ ∞ := sup [ π ] ∥ A ( π ) ∥ op < ∞ } ,

where ℋ π is the Hilbert space of π. The Fourier transform ℱ of 𝒜 is defined by

ℱ ⁢ ( a ) := a ^ := ( π ⁢ ( a ) ) [ π ] ∈ 𝒜 ^ for ⁢ a ∈ 𝒜 .

It is then an injective, and hence isometric, homomorphism from 𝒜 into l ∞ ⁢ ( 𝒜 ^ ) . Therefore, we can analyze the C*-algebra 𝒜 by recognizing 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 [9] 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, 8]. In serial, the C*-algebra of a ⁢ x + 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 in [11], have all been characterized 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 [2, 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 ) i = 0 d , 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 a ^ , a ∈ C * ⁢ ( G ) , in particular the behavior of the operators a ^ ⁢ ( 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 behavior of the operator fields a ^ , a ∈ C * ⁢ ( G ) , is more involved.

We first recall the following definitions, which were given in [1]. Let ℋ be a Hilbert space and let ℬ ⁢ ( ℋ ) denote the algebra of bounded linear operators on ℋ .

Definition 1.1.

Let d be a natural number.

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.
Let S be locally compact of step ≤ d and let { 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 ∩ Γ i , 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

∥ ψ ∥ M = sup ⁡ { ∥ ψ ⁢ ( γ ) ∥ ℬ ⁢ ( ℋ i ) : M ∩ Γ i ≠ ∅ , γ ∈ M ∩ Γ i } .
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

l ∞ ⁢ ( S ) := { ( ϕ ⁢ ( s ) ) s ∈ S : ϕ ⁢ ( s ) ∈ ℬ ⁢ ( ℋ s ) , s ∈ S , sup s ∈ S ⁡ ∥ ϕ ⁢ ( s ) ∥ op < ∞ } .

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

∅ = S - 1 ⊂ S 0 ⊂ S 1 ⊂ ⋯ ⊂ S d = 𝒜 ^ .

Assume that, for every i ∈ { 0 , … , d } , there exist a Hilbert space ℋ i and a concrete realization ( 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 in Definition 1.1 (iii), i.e. for a ∈ 𝒜 , let

ℱ ⁢ ( a ) ⁢ ( γ ) = a ^ ⁢ ( γ ) := π γ ⁢ ( a ) for ⁢ γ ∈ Γ i ⁢ and ⁢ i = 0 , … , d .

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 realization ( π γ , ℋ i ) of γ on the Hilbert space ℋ i such that ℱ ⁢ ( 𝒜 ) | Γ i is contained in CB ⁡ ( Γ i , ℋ i ) for every 0 ≤ i ≤ d .

2 The Boidol group

2.1 Definition

Let 𝔤 be the real Lie algebra of dimension 4 with a basis { T , X , Y , Z } and the non-trivial brackets

[ T , X ] = - X , [ T , Y ] = Y , [ X , Y ] = Z .

The connected simply connected group G with Lie algebra 𝔤 , which we call the Boidol group, can be realized on ℝ 4 with the multiplication

( t , x , y , z ) ⋅ ( t ′ , x ′ , y ′ , z ′ ) = ( t + t ′ , e t ′ ⁢ x + x ′ , e - t ′ ⁢ y + y ′ , z + z ′ + 1 2 ⁢ ( e t ′ ⁢ x ⁢ y ′ - e - t ′ ⁢ x ′ ⁢ y ) ) .

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 center of G. Furthermore,

G = ℝ ⋉ H

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 * .

The orbits in general position: For ( ρ , λ ) ∈ ℝ × ℝ * , where ℝ * = ℝ ∖ { 0 } , let ℓ = ℓ ρ , λ = ( ρ , 0 , 0 , λ ) ∈ 𝔤 * . The stabilizer G ⁢ ( ℓ ) of ℓ in G is given by G ⁢ ( ℓ ) = exp ⁡ ℝ ⁢ T ⋅ exp ⁡ ℝ ⁢ Z , and the coadjoint orbit O ρ , λ of ℓ ρ , λ is the subset

O ρ , λ = { ( ρ ⁢ λ + x ⁢ y λ ) ⁢ T * + x ⁢ X * + y ⁢ Y * + λ ⁢ Z * : x , y ∈ ℝ }

of 𝔤 * .
The orbits of dimension 2 vanishing on Z: For ( α , β ) ∈ ℝ 2 with α 2 + β 2 ≠ 0 , the coadjoint orbit

O α , β , 0 = Ad * ⁡ ( G ) ⁢ ℓ α , β , 0

of the element ℓ α , β , 0 = α ⁢ X * + β ⁢ Y * is given by

O α , β , 0 = { u ⁢ T * + ( e t ⁢ α ) ⁢ X * + ( e - t ⁢ β ) ⁢ Y * : t , u ∈ ℝ }

= { u ⁢ T * + ( | β | ⁢ α ⁢ e t ) ⁢ X * + ( sign ⁡ ( β ) ⁢ e - t ) ⁢ Y * : t , u ∈ ℝ }

= { u ⁢ T * + ( sign ⁡ ( α ) ⁢ e t ) ⁢ X * + ( | α | ⁢ β ⁢ e - t ) ⁢ Y * : t , u ∈ ℝ } .
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 * :

𝔤 * = Γ 3 ∪ Γ 2 ∪ Γ 1 ∪ Γ 0 ,

where the following properties are satisfied:

Γ 3 = { O ρ , λ : ρ ∈ ℝ , λ ∈ ℝ * } . The subset

Σ 3 := { ℓ ρ , λ : ( ρ , λ ) ∈ ℝ × ℝ * }

is a section for the G-orbits in Γ 3 .
Let

Γ 2 , ε , σ := { O ρ ⁢ ε , σ , 0 : ρ ∈ ℝ + , ε , σ ∈ { + 1 , - 1 } }

and

Γ 2 := ⋃ ε , σ ∈ { + 1 , - 1 } Γ 2 , ε , σ .

For ω ∈ ℝ * and σ ∈ { + 1 , - 1 } , let

ℓ ω , σ , 0 := ( 0 , ω , σ , 0 ) ∈ O ω , σ , 0 .

The subset Σ 2 of Γ 2 , defined by

Σ 2 := { ℓ ω , σ , 0 : σ ∈ { + 1 , - 1 } , ω ∈ ℝ * } ,

is a section for the G-orbits in Γ 2 .
Γ 1 := { O 1 , 0 , 0 , O - 1 , 0 , 0 , O 0 , 1 , 0 , O 0 , - 1 , 0 } . The subset Σ 1 of Γ 1 , given by

Σ 1 := { ℓ σ , 0 , 0 , ℓ 0 , σ , 0 : σ ∈ { + 1 , - 1 } } ,

is a section for the G-orbits in Γ 1 .
Finally, let

Σ 0 = Γ 0 := { ℓ τ : τ ∈ ℝ } ,

where ℓ τ := ( τ , 0 , 0 , 0 ) .

We recall that

S i = ⋃ j = 0 i Γ j , i = 0 , 1 , 2 , 3 .

Remark 2.1.

The spectrum of G / 𝒵 (and of C * ⁢ ( G / 𝒵 ) ) can be identified with the closed subset

{ π ∈ G ^ : π ⁢ ( 𝒵 ) = { 𝕀 ℋ π } }

and also with the subset

S 2 = { Ω ∈ 𝔤 * / G : ℓ ⁢ ( Z ) = 0 , ℓ ∈ Ω } = Γ 2 ∪ Γ 1 ∪ Γ 0

of the coadjoint orbit space.
Let ( λ G / 𝒵 , L 2 ⁢ ( G / 𝒵 ) ) be the left regular representation of the group G / 𝒵 and let ( λ ~ G / 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 [13]) that

λ ~ G / 𝒵 ⁢ ( C * ⁢ ( G ) ) = λ G / 𝒵 ⁢ ( C * ⁢ ( G / 𝒵 ) ) .

Furthermore, the kernel of λ ~ G / 𝒵 in C * ⁢ ( G ) is the ideal

K S 2 = { a ∈ C * ⁢ ( G ) : π ⁢ ( a ) = 0 ⁢ for ⁢ π ∈ S 2 } .

The C*-algebra C * ⁢ ( G / 𝒵 ) is thus isomorphic to the quotient of C * ⁢ ( G ) by K S 2 .
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].
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 1 are 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 π Ω ) on the Hilbert space ℋ π Ω of π Ω .

Proposition 2.2.

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 [9, p. 135]). ∎

In the following, we recall the definition of properly converging sequences and their limit sets, and 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, is equal 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.

Proposition 2.4.

Let O ¯ := ( O ρ k , λ k ) k be a properly converging sequence in Γ 3 such that lim k → ∞ ⁡ λ k = 0 .

The sequence ( ω k ) k := ( ρ k ⁢ λ k ) k converges to some ω ∈ ℝ .
If ω ≠ 0 , then the limit set L of the sequence O ¯ is the two point set { O ω , - 1 , 0 , O - ω , 1 , 0 } ⊂ Γ 2 .
If ω = 0 , then the limit set L is the subset Γ 1 ∪ Γ 0 of the orbit space.

Proof.

(i) Suppose that the sequence ( O ρ k , λ k ) k ⊂ Γ 3 converges to some O ∈ Γ 2 ∪ Γ 1 ∪ Γ 0 . Take

ℓ = t ⁢ T * + x ⁢ X * + y ⁢ Y * ∈ O .

Then, for any k ∈ ℕ , we have an element

ℓ k = ( ρ k ⁢ λ k + x k ⁢ y k λ k ) ⁢ T * + x k ⁢ X * + y k ⁢ Y * + λ k ⁢ Z * ∈ O ρ k , λ k

which converges to ℓ . In particular,

lim k → ∞ ⁡ y k = y , lim k → ∞ ⁡ x k = x , lim k → ∞ ⁡ ρ k ⁢ λ k + x k ⁢ y k λ k = t .

Let ω := - x ⁢ y . Since lim k → ∞ ⁡ λ k = 0 , it follows that lim k → ∞ ⁡ ρ k ⁢ λ k = ω .

(ii) Suppose that ω ≠ 0 . Let O be a limit point of the sequence ( O ρ k , λ k ) k . Then O is contained in Γ 2 ∪ Γ 1 ∪ Γ 0 . Let

ℓ = t ⁢ T * + x ⁢ X * + y ⁢ Y * ∈ O .

Since 0 ≠ ω = - x ⁢ y , we have that O ∈ Γ 2 . We can assume that t = 0 , | x | = | ω | and | y | = 1 . If y = 1 , then x = - ω , and conversely if y = - 1 , then x = ω .

(iii) If ω = 0 , then lim k → ∞ ⁡ x k ⁢ y k = 0 . Take ℓ = σ ⁢ X * ∈ Γ 1 . We can use

ℓ k = σ ⁢ X * - σ ⁢ ω k ⁢ Y * + λ k ⁢ Z * = ( ρ k ⁢ λ k + σ ⁢ ( - σ ⁢ ω k ) λ k ) ⁢ T * + σ ⁢ X * - σ ⁢ ω k ⁢ Y * + λ k ⁢ Z * ∈ O ρ k , λ k

and ω k := ρ k ⁢ λ k for k ∈ ℕ . Then lim k → ∞ ⁡ ℓ k = ℓ . Similarly, for ℓ = σ ⁢ Y * . For ℓ = α ⁢ T * ∈ Γ 0 , we take

x k := - | α ⁢ λ k - ω k | and y k := | α ⁢ λ k - ω k | ⁢ sign ⁡ ( α ⁢ λ k - ω k ) ,

provided α ⁢ λ k - ω k ≠ 0 . Otherwise, we use x k := 0 = : y k . Then

ℓ k = α ⁢ T * + λ k ⁢ Z * ∈ O ρ k , λ k and lim k → ∞ ⁡ ℓ k = ℓ .

This concludes the proof. ∎

The following proposition can be found in [10, Theorem 2.3].

Proposition 2.5.

Let O ¯ = ( O ε ⁢ ρ k , σ , 0 ) k , with ρ k > 0 and ε , σ ∈ { + 1 , - 1 } , be a properly converging sequence in Γ 2 with limits in Γ 0 ∪ Γ 1 .

lim k → ∞ ⁡ ρ k = 0 and the limit set of O ¯ is { O ε , 0 , 0 , O 0 , σ , 0 } ∪ Γ 0 .
The closure of O σ , 0 , 0 (resp. O 0 , σ , 0 ) is O σ , 0 , 0 ∪ Γ 0 (resp. O 0 , σ , 0 ∪ Γ 0 ) for σ = + 1 or σ = - 1 .

Note that, throughout the paper, the symbols σ , ε will denote elements of the set { + 1 , - 1 } .

3 The Fourier transform

Definition 3.1.

Let ( π , ℋ ) be a representation of G and let α : G → G be an automorphism. We define the new representation ( π α , ℋ ) of G by

π α ⁢ ( g ) := π ⁢ ( α - 1 ⁢ ( g ) ) for ⁢ g ∈ G .

3.1 The irreducible representations

Let the automorphism α : G → G be given by

α ⁢ ( t , x , y , z ) := ( t , - x , - y , z ) .

(I) For ℓ ρ , λ ∈ Γ 3 , consider the Pukanszky polarization 𝔭 := span ⁡ { T , Y , Z } at ℓ ρ , λ . This gives us the irreducible representation

π ρ , λ := ind P G ⁡ χ ρ , λ ,

where P = exp ⁡ 𝔭 and

χ ρ , λ ⁢ ( exp ⁡ ( t ⁢ T ) ⁢ exp ⁡ ( y ⁢ Y ) ⁢ exp ⁡ ( z ⁢ Z ) ) := e - i ⁢ ( ρ ⁢ t + λ ⁢ z ) 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 realized as L 2 ⁢ ( ℝ ) . Let ( t , x , y , z ) ∈ G , ξ ∈ L 2 ⁢ ( ℝ ) and u ∈ ℝ . We have

π ρ , λ ⁢ ( t , x , y , z ) ⁢ ξ ⁢ ( u ) = e t / 2 ⁢ e - i ⁢ ρ ⁢ t ⁢ e - i ⁢ λ ⁢ z ⁢ e - i ⁢ λ ⁢ x ⁢ y / 2 ⁢ e i ⁢ λ ⁢ e t ⁢ y ⁢ u ⁢ ξ ⁢ ( e t ⁢ u - x ) .

For F ∈ L 1 ⁢ ( G ) , define its partial Fourier transform F ^ 3 , 4 by

F ^ 3 , 4 ⁢ ( t , x , a , b ) := ∫ ℝ 2 F ⁢ ( t , x , y , z ) ⁢ e - i ⁢ ( a ⁢ y + b ⁢ z ) ⁢ 𝑑 y ⁢ 𝑑 z for all ⁢ t , x , a , b ∈ ℝ .

Then, for any F ∈ L 1 ⁢ ( G ) such that F ^ 3 , 4 is contained in C c ⁢ ( ℝ 4 ) , ξ ∈ L 2 ⁢ ( ℝ ) and u ∈ ℝ , we have that

(3.1)

π ρ , λ ⁢ ( F ) ⁢ ξ ⁢ ( u ) = ∫ G F ⁢ ( g ) ⁢ π ρ , λ ⁢ ( g ) ⁢ ξ ⁢ ( u ) ⁢ 𝑑 g

= ∫ ℝ ( ∫ ℝ e t / 2 ⁢ e - i ⁢ ρ ⁢ t ⁢ F ^ 3 , 4 ⁢ ( t , x , λ ⁢ ( x 2 - e t ⁢ u ) , λ ) ⁢ ξ ⁢ ( e t ⁢ u - x ) ⁢ 𝑑 t ) ⁢ 𝑑 x

= ∫ ℝ ( ∫ ℝ e t / 2 ⁢ e - i ⁢ ρ ⁢ t ⁢ F ^ 3 , 4 ⁢ ( t , e t ⁢ u - x , - λ 2 ⁢ ( x + e t ⁢ u ) , λ ) ⁢ 𝑑 t ) ⁢ ξ ⁢ ( x ) ⁢ 𝑑 x .

Therefore, π ρ , λ is a kernel operator with kernel function

(3.2) F ρ , λ ⁢ ( u , x ) = ∫ ℝ e t / 2 ⁢ e - i ⁢ ρ ⁢ t ⁢ F ^ 3 , 4 ⁢ ( t , e t ⁢ u - x , - λ 2 ⁢ ( x + e t ⁢ u ) , λ ) ⁢ 𝑑 t for ⁢ u , x ∈ ℝ .

Furthermore, for u ∈ ℝ , we obtain the identity

π ρ , λ α ⁢ ( t , x , y , z ) ⁢ ξ ⁢ ( u ) = e t / 2 ⁢ e - i ⁢ ρ ⁢ t ⁢ e - i ⁢ λ ⁢ z ⁢ e - i ⁢ λ ⁢ x ⁢ y / 2 ⁢ e - i ⁢ λ ⁢ e t ⁢ y ⁢ u ⁢ ξ ⁢ ( e t ⁢ u + x ) ,

which shows that

(3.3)

π ρ , λ α ⁢ ( F ) ⁢ ξ ⁢ ( u ) = ∫ R 4 e t / 2 ⁢ e - i ⁢ ρ ⁢ t ⁢ e - i ⁢ λ ⁢ z ⁢ e - i ⁢ λ ⁢ x ⁢ y / 2 ⁢ e - i ⁢ λ ⁢ e t ⁢ y ⁢ u ⁢ F ⁢ ( t , x , y , z ) ⁢ ξ ⁢ ( e t ⁢ u + x ) ⁢ 𝑑 z ⁢ 𝑑 y ⁢ 𝑑 x ⁢ 𝑑 t

= ∫ ℝ ( ∫ ℝ e t / 2 ⁢ e - i ⁢ ρ ⁢ t ⁢ F ^ 3 , 4 ⁢ ( t , - e t ⁢ u + x , λ 2 ⁢ ( x + e t ⁢ u ) , λ ) ⁢ 𝑑 t ) ⁢ ξ ⁢ ( x ) ⁢ 𝑑 x .

Let S : L 2 ⁢ ( ℝ ) → L 2 ⁢ ( ℝ ) be the unitary operator defined by

S ⁢ ( ξ ) ⁢ ( u ) := ξ ⁢ ( - u ) for all ⁢ ξ ∈ L 2 ⁢ ( ℝ ) , u ∈ ℝ .

Then

(3.4)

S ∘ π ρ , λ ⁢ ( t , x , y , z ) ∘ S ⁢ ( ξ ) ⁢ ( u ) = e t / 2 ⁢ e - i ⁢ ρ ⁢ t ⁢ e - i ⁢ λ ⁢ z ⁢ e - i ⁢ λ ⁢ x ⁢ y / 2 ⁢ e - i ⁢ λ ⁢ e t ⁢ y ⁢ u ⁢ ξ ⁢ ( e t ⁢ u + x )

= π ρ , λ α ⁢ ( t , x , y , z ) ⁢ ( ξ ) ⁢ ( u ) , ( t , x , y , z ) ∈ G .

Later in Section 4, we shall need an equivalent version of the representations π ρ , λ .

Definition 3.2.

For any ( ρ , λ ) ∈ ℝ × ℝ * and any measurable function φ : ℝ → ℝ , the operator

V : L 2 ⁢ ( ℝ , d ⁢ x | x | ) → L 2 ⁢ ( ℝ , d ⁢ x )

defined by

V ⁢ ( η ) ⁢ ( s ) = ξ ⁢ ( s ) := 1 | s | 1 / 2 ⁢ η ⁢ ( | λ | ⁢ s ) ⁢ e i ⁢ φ ⁢ ( λ ⁢ s ) , η ∈ L 2 ⁢ ( ℝ , d ⁢ x | x | ) , s ∈ ℝ ,

is a unitary operator. We have that

V * ⁢ ( ξ ) ⁢ ( u ) = | u | 1 / 2 | λ | ⁢ e - i ⁢ φ ⁢ ( u ) ⁢ ξ ⁢ ( u | λ | ) , u ∈ ℝ , ξ ∈ L 2 ⁢ ( ℝ ) .

We can also view V as an operator

V : L 2 ⁢ ( ℝ + , d ⁢ x | x | ) ⊕ L 2 ⁢ ( ℝ - , d ⁢ x | x | ) → L 2 ⁢ ( ℝ + , d ⁢ x ) ⊕ L 2 ⁢ ( ℝ - , d ⁢ x ) .

It is easy to see that

(3.5) S ∘ V = V ∘ S ,

where

S : L 2 ⁢ ( ℝ σ , d ⁢ u | u | ) → L 2 ⁢ ( ℝ - σ , d ⁢ u | u | )

is defined as before by

S ⁢ ( η ) ⁢ ( u ) = η ⁢ ( - u ) , u ∈ ℝ σ .

Let us now take the measurable function φ as

φ ⁢ ( s ) := e i ⁢ ρ ⁢ ln ⁡ ( | s | ) , s ∈ ℝ .

For η ∈ C c ∞ ⁢ ( ℝ + ) , s ∈ ℝ and F ^ 3 , 4 ∈ C c ∞ ⁢ ( ℝ 4 ) ⊂ L 1 ⁢ ( G ) , we see by (3.1) that

(3.6) π ρ , λ ⁢ ( F ) ⁢ ( V ⁢ ( η ) ) ⁢ ( s ) = ∫ ℝ ∫ ℝ e t / 2 ⁢ e - i ⁢ ρ ⁢ t ⁢ F ^ 3 , 4 ⁢ ( t , x , - λ 2 ⁢ ( - x + 2 ⁢ e t ⁢ s ) , λ ) ⁢ e i ⁢ ρ ⁢ ln ⁡ ( | - λ ⁢ x + λ ⁢ e t ⁢ s | ) ⁢ | λ | 1 / 2 | - λ ⁢ x + e t ⁢ λ ⁢ s | 1 / 2 ⁢ η ⁢ ( - | λ | ⁢ x + | λ | ⁢ e t ⁢ s ) ⁢ 𝑑 x ⁢ 𝑑 t .

Hence, writing λ = ε ⁢ | λ | , we obtain

(3.7)

V * ( π ρ , λ ( F ) ( V ( η ) ) ) ( s )

= e - i ⁢ ρ ⁢ ln ⁡ ( | s | ) ⁢ ∫ ℝ ∫ ℝ e - i ⁢ ρ ⁢ t ⁢ F ^ 3 , 4 ⁢ ( t , x , λ 2 ⁢ x - ε ⁢ e t ⁢ s , λ ) ⁢ e i ⁢ ρ ⁢ ln ⁡ ( | - λ ⁢ x + ε ⁢ e t ⁢ s | ) ⁢ | e t ⁢ s | 1 / 2 | - ε ⁢ λ ⁢ x + e t ⁢ s | 1 / 2 ⁢ η ⁢ ( - ε ⁢ λ ⁢ x + e t ⁢ s ) ⁢ 𝑑 x ⁢ 𝑑 t

= ∫ ℝ ∫ ℝ F ^ 3 , 4 ⁢ ( t , x , λ 2 ⁢ x - ε ⁢ e t ⁢ s , λ ) ⁢ e i ⁢ ρ ⁢ ( ln ⁡ ( | - ε ⁢ λ ⁢ x + e t ⁢ s | - ln ⁡ ( | e t ⁢ s | ) ) ) ⁢ | e t ⁢ s | 1 / 2 | - ε ⁢ λ ⁢ x + e t ⁢ s | 1 / 2 ⁢ η ⁢ ( - ε ⁢ λ ⁢ x + e t ⁢ s ) ⁢ 𝑑 x ⁢ 𝑑 t

= e - i ⁢ ρ ⁢ ln ⁡ ( | s | ) ⁢ ∫ ℝ ∫ ℝ e - i ⁢ ρ ⁢ t ⁢ F ^ 3 , 4 ⁢ ( t , 1 λ ⁢ ( λ ⁢ x + ε ⁢ e t ⁢ s ) , 1 2 ⁢ ( λ ⁢ x - ε ⁢ e t ⁢ s ) , λ ) ⁢ e i ⁢ ρ ⁢ ln ⁡ ( | λ ⁢ x | ) ⁢ | e t ⁢ s | 1 / 2 | λ ⁢ x | 1 / 2 ⁢ η ⁢ ( - ε ⁢ λ ⁢ x ) ⁢ 𝑑 x ⁢ 𝑑 t

= e - i ⁢ ρ ⁢ ln ⁡ ( | s | ) ⁢ | s | 1 / 2 | λ | ⁢ ∫ ℝ ∫ ℝ e - i ⁢ ρ ⁢ t ⁢ e t / 2 ⁢ F ^ 3 , 4 ⁢ ( t , - ε λ ⁢ ( x - e t ⁢ s ) , - ε 2 ⁢ ( x + e t ⁢ s ) , λ ) ⁢ 𝑑 t ⁢ e i ⁢ ρ ⁢ ln ⁡ ( | x | ) | x | 1 / 2 ⁢ η ⁢ ( x ) ⁢ 𝑑 x

= e - i ⁢ ρ ⁢ ln ⁡ ( | s | ) ⁢ | s | 1 / 2 | λ | ⁢ ∫ ℝ H ⁢ ( s , x ) ⁢ e i ⁢ ρ ⁢ ln ⁡ ( | x | ) | x | 1 / 2 ⁢ η ⁢ ( x ) ⁢ 𝑑 x ,

where

H ⁢ ( s , x ) := ∫ ℝ e - i ⁢ ρ ⁢ t ⁢ e t / 2 ⁢ F ^ 3 , 4 ⁢ ( t , - ε λ ⁢ ( x - e t ⁢ s ) , - ε 2 ⁢ ( x + e t ⁢ s ) , λ ) ⁢ 𝑑 t , x , s ∈ ℝ .

Using partial integration, we see that

H ⁢ ( s , x ) = 1 i ⁢ ρ ⁢ ∫ ℝ e - i ⁢ ρ ⁢ t ⁢ ∂ t ⁡ ( e t / 2 ⁢ F ^ 3 , 4 ⁢ ( t , - ε λ ⁢ ( x - e t ⁢ s ) , - ε 2 ⁢ ( x + e t ⁢ s ) , λ ) ) ⁡ d ⁢ t

= 1 i ⁢ ρ ∫ ℝ e - i ⁢ ρ ⁢ t ( 1 2 e t / 2 F ^ 3 , 4 ( t , - ε λ ( x - e t s ) , - ε 2 ( x + e t s ) , λ ) + e t / 2 ε e t ⁢ s λ ∂ 2 F ^ 3 , 4 ( t , - ε λ ( x - e t s ) , - ε 2 ( x + e t s ) , λ )

- e t / 2 ε 2 e t s ∂ 3 F ^ 3 , 4 ( t , - ε λ ( x - e t s ) , - ε 2 ( x + e t s ) , λ ) ) d t .

Hence, there exist positive continuous functions ψ , ψ ′ with compact support on ℝ such that

(3.8)

| H ⁢ ( s , x ) | ≤ 1 | ρ ⁢ λ | ⁢ ψ ′ ⁢ ( λ ) ⁢ ( 1 + | s | ) ⁢ ∫ ℝ ψ ′ ⁢ ( t ) ⁢ ψ ′ ⁢ ( x - e t ⁢ s λ ) ⁢ ψ ′ ⁢ ( x + e t ⁢ s ) ⁢ 𝑑 t

≤ 1 | ρ ⁢ λ | ⁢ ψ ⁢ ( λ ) ⁢ ∫ ℝ ψ ⁢ ( t ) ⁢ ψ ⁢ ( x - e t ⁢ s λ ) ⁢ ψ ⁢ ( x + e t ⁢ s ) ⁢ 𝑑 t , s , x ∈ ℝ .

Furthermore, with the automorphism α on the group G and s ∈ ℝ , we have that

(3.9)

V * ⁢ ( π ρ , λ α ⁢ ( F ) ⁢ ( V ⁢ ( η ) ) ) ⁢ ( s ) = e - i ⁢ ρ ⁢ ln ⁡ ( | s | ) ⁢ ∫ ℝ ∫ ℝ e - i ⁢ ρ ⁢ t ⁢ F ^ 3 , 4 ⁢ ( t , x , λ 2 ⁢ x + ε ⁢ e t ⁢ s , λ ) ⁢ e i ⁢ ρ ⁢ ln ⁡ ( | λ ⁢ x + ε ⁢ e t ⁢ s | ) ⁢ | e t ⁢ s | 1 / 2 | ε ⁢ λ ⁢ x + e t ⁢ s | 1 / 2 ⁢ η ⁢ ( ε ⁢ λ ⁢ x + e t ⁢ s ) ⁢ 𝑑 x ⁢ 𝑑 t

= ∫ ℝ ∫ ℝ F ^ 3 , 4 ⁢ ( t , x , λ 2 ⁢ x + ε ⁢ e t ⁢ s , λ ) ⁢ e i ρ ( ln ( | λ x + ε e t s | - ln ( | e t s | ) ) ⁢ | e t ⁢ s | 1 / 2 | λ ⁢ x + ε ⁢ e t ⁢ s | 1 / 2 ⁢ η ⁢ ( ε ⁢ λ ⁢ x + e t ⁢ s ) ⁢ 𝑑 x ⁢ 𝑑 t

= ∫ ℝ ∫ ℝ F ^ 3 , 4 ⁢ ( t , - x , - λ 2 ⁢ x + ε ⁢ e t ⁢ s , λ ) ⁢ e i ρ ( ln ( | - λ x + ε e t s | - ln ( | e t s | ) ) ⁢ | e t ⁢ s | 1 / 2 | - λ ⁢ x + ε ⁢ e t ⁢ s | 1 / 2 ⁢ η ⁢ ( - ε ⁢ λ ⁢ x + e t ⁢ s ) ⁢ 𝑑 x ⁢ 𝑑 t .

(II) For ℓ ∈ Γ 2 ∪ Γ 1 , the subalgebra 𝔥 := span ⁡ { X , Y , Z } is a Pukanszky polarization at ℓ . Therefore, the unitary representation

π ℓ := ind H G ⁡ χ ℓ ,

where H := exp ⁡ 𝔥 and χ ℓ ⁢ ( exp ⁡ ( U ) ) := e - i ⁢ ℓ ⁢ ( U ) for U ∈ 𝔤 , is irreducible.

Take ℓ := ( 0 , μ , ν , 0 ) for some μ , ν ∈ ℝ with μ 2 + ν 2 ≠ 0 . We have that

L 2 ⁢ ( G / H , χ ℓ ) ≃ L 2 ⁢ ( ℝ )

and, for ( t , x , y , z ) ∈ G , φ ∈ L 2 ⁢ ( ℝ ) and v ∈ ℝ ,

π ℓ ⁢ ( t , x , y , z ) ⁢ φ ⁢ ( v ) = e - i ⁢ μ ⁢ e v - t ⁢ x ⁢ e - i ⁢ ν ⁢ e t - v ⁢ y ⁢ φ ⁢ ( v - t ) .

Hence for F ∈ L 1 ⁢ ( G ) ,

π ℓ ⁢ ( F ) ⁢ φ ⁢ ( v ) = ∫ ℝ F ^ 2 , 3 , 4 ⁢ ( v - t , μ ⁢ e t , ν ⁢ e - t , 0 ) ⁢ φ ⁢ ( t ) ⁢ 𝑑 t .

This means that π ℓ ⁢ ( F ) is a kernel operator with kernel function

(3.10) F ℓ ⁢ ( u , t ) = F ^ 2 , 3 , 4 ⁢ ( v - t , μ ⁢ e t , ν ⁢ e - t , 0 ) for ⁢ v , t ∈ ℝ .

Let us write an equivalent representation for π ℓ : We use the multiplication invariant measure d ⁢ u | u | on ℝ σ , σ ∈ { + 1 , - 1 } . Let

U σ : L 2 ⁢ ( ℝ ) → L 2 ⁢ ( ℝ σ , d ⁢ u | u | ) be defined by U σ ⁢ ξ ⁢ ( u ) := ξ ⁢ ( - ln ⁡ ( σ ⁢ u ) ) , u ∈ ℝ σ .

Then U σ is a unitary operator and

( U σ ) * ⁢ ( η ) ⁢ ( s ) = η ⁢ ( σ ⁢ e - s ) for ⁢ s ∈ ℝ , η ∈ L 2 ⁢ ( ℝ σ , d ⁢ u | u | ) .

Let

(3.11) τ μ , ν σ := U σ ∘ π ( 0 , μ , ν , 0 ) ∘ U σ * .

We obtain the relation

τ μ , ν σ ⁢ ( t , x , y , z ) ⁢ η ⁢ ( u ) = ( π ℓ ⁢ ( t , x , y , z ) ⁢ ( U σ * ⁢ ( η ) ) ) ⁢ ( - ln ⁡ ( σ ⁢ u ) )

= e - i ⁢ μ ⁢ e - ln ⁡ ( σ ⁢ u ) - t ⁢ x ⁢ e - i ⁢ ν ⁢ e t + ln ⁡ ( σ ⁢ u ) ⁢ y ⁢ U σ * ⁢ ( η ) ⁢ ( - ln ⁡ ( σ ⁢ u ) - t )

= e - i ⁢ ( σ ⁢ μ ) ⁢ u - 1 ⁢ e - t ⁢ x ⁢ e - i ⁢ ( σ ⁢ ν ) ⁢ u ⁢ e t ⁢ y ⁢ η ⁢ ( e t ⁢ u ) , η ∈ L 2 ⁢ ( ℝ σ , d ⁢ u | u | ) .

We see that

(3.12)

( τ μ , ν σ ) ( t , x , y , z ) α η ( u ) = e i ⁢ ( σ ⁢ μ ) ⁢ u - 1 ⁢ e - t ⁢ x e i ⁢ ( σ ⁢ ν ) ⁢ u ⁢ e t ⁢ y η ( e t u )

= ( τ - μ , - ν σ ) ⁢ ( t , x , y , z ) ⁢ η ⁢ ( u ) , η ∈ L 2 ⁢ ( ℝ σ , d ⁢ u | u | ) .

On the other hand,

(3.13)

S ∘ τ μ , ν σ ⁢ ( t , x , y , z ) ∘ S ⁢ ( η ) ⁢ ( u ) = e - i ⁢ ( - σ ⁢ μ ) ⁢ u - 1 ⁢ e - t ⁢ x ⁢ e - i ⁢ ( - σ ⁢ ν ) ⁢ u ⁢ e t ⁢ y ⁢ η ⁢ ( e t ⁢ u )

= τ μ , ν - σ ⁢ ( t , x , y , z ) ⁢ η ⁢ ( u ) , η ∈ L 2 ⁢ ( ℝ - σ , d ⁢ u | u | ) , u ∈ ℝ - σ .

In particular, the representations τ μ , ν σ and τ μ , ν - σ are equivalent. Furthermore, for η ∈ L 2 ⁢ ( ℝ σ , d ⁢ u | u | ) and F ∈ L 1 ⁢ ( G ) ,

(3.14)

τ μ , ν σ ⁢ ( F ) ⁢ η ⁢ ( u ) = ∫ G F ⁢ ( t , x , y , z ) ⁢ e - i ⁢ σ ⁢ μ ⁢ u - 1 ⁢ e - t ⁢ x ⁢ e - i ⁢ σ ⁢ ν ⁢ u ⁢ e t ⁢ y ⁢ η ⁢ ( e t ⁢ u ) ⁢ 𝑑 t ⁢ 𝑑 x ⁢ 𝑑 y ⁢ 𝑑 z

= ∫ ℝ F ^ 2 , 3 , 4 ⁢ ( t , σ ⁢ μ ⁢ ( e t ⁢ u ) - 1 , σ ⁢ ν ⁢ ( e t ⁢ u ) , 0 ) ⁢ η ⁢ ( e t ⁢ u ) ⁢ 𝑑 t

= ∫ ℝ F ^ 2 , 3 , 4 ⁢ ( t - ln ⁡ ( σ ⁢ u ) , μ ⁢ e - t , ν ⁢ e t , 0 ) ⁢ η ⁢ ( σ ⁢ e t ) ⁢ 𝑑 t .

(III) For ℓ ∈ Γ 0 , let

π ℓ := χ ℓ .

Hence for F ∈ L 1 ⁢ ( G ) ,

π ℓ ⁢ ( F ) = ∫ ℝ ∫ ℝ 3 e - i ⁢ t ⁢ ℓ ⁢ F ⁢ ( t , h ) ⁢ 𝑑 h ⁢ 𝑑 t .

In order to define the Fourier transform a ↦ a ^ , a ∈ C * ⁢ ( G ) , we identify G ^ with the set Γ := Γ 3 ∪ Γ 2 ∪ Γ 1 ∪ Γ 0 and we let

(3.15) a ^ ⁢ ( ρ , λ ) := π ρ , λ ⁢ ( a ) ∈ 𝒦 ⁢ ( L 2 ⁢ ( ℝ ) ) , ρ ∈ ℝ , λ ∈ ℝ * ,

a ^ ⁢ ( μ , ν , 0 ) := τ μ , ν + ⁢ ( a ) ∈ 𝒦 ⁢ ( L 2 ⁢ ( ℝ + , d ⁢ u u ) ) , ℓ μ , ν , 0 ∈ Γ 2 ,

a ^ ⁢ ( μ , ν , 0 ) := τ μ , ν + ⁢ ( a ) ∈ ℬ ⁢ ( L 2 ⁢ ( ℝ + , d ⁢ u u ) ) , ℓ μ , ν , 0 ∈ Γ 1 ,

a ^ ⁢ ( ℓ ) := χ ℓ ⁢ ( a ) ∈ ℂ , ℓ ∈ Γ 0 .

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 characterize the C*-algebra of the Boidol group which, will be our main result (Theorem 4.24).

4.1 Norm convergence

Definition 4.1.

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.

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 .

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 ) . Expression (3.1) tells us that the operators π ℓ ⁢ ( F ) , ℓ ∈ Γ 3 , are Hilbert–Schmidt operators. 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 lim k → ∞ ⁡ O k = ∞ in Γ 3 , then, according to Proposition 2.4, we have that lim k → ∞ ⁡ ρ k ⁢ λ k = ∞ . If lim k → ∞ ⁡ λ k = ∞ , then π ρ k , λ k ⁢ ( F ) = 0 for k large enough, and if lim k → ∞ ⁡ ρ k = ∞ , then

| F ρ k , λ k ⁢ ( x , u ) | ≤ 1 | ρ k | ⁢ ψ ⁢ ( x - u )

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]. ∎

4.2 An approximation of π ρ k , λ k ⁢ ( F )

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 can assume (by passing to a subsequence if necessary) that λ k = ε ⁢ | λ k | for every k ∈ ℕ .

Definition 4.3.

Let c ≤ d ∈ [ - ∞ , ∞ ] and δ > 0 . We define a family of multiplication operators on the space L 2 ⁢ ( ℝ ) or L 2 ⁢ ( ℝ ± , d ⁢ x | x | ) : for s ∈ ℝ and ξ ∈ L 2 ⁢ ( ℝ ) (or L 2 ⁢ ( ℝ ± , d ⁢ x | x | ) ),

M { c ≤ d } ⁢ ( ξ ) ⁢ ( s ) := 1 [ c , d ] ⁢ ( s ) ⁢ ξ ⁢ ( s ) ,

M { ≤ d } ⁢ ( ξ ) ⁢ ( s ) := 1 [ - ∞ , d ] ⁢ ( s ) ⁢ ξ ⁢ ( s ) ,

M { d ≤ } ⁢ ( ξ ) ⁢ ( s ) := 1 [ d , ∞ ] ⁢ ξ ⁢ ( s ) ,

M { | | ≤ δ } ⁢ ( ξ ) ⁢ ( s ) := 1 [ - δ , δ ] ⁢ ( s ) ⁢ ξ ⁢ ( s ) ,

M { δ ≤ | | } ⁢ ( ξ ) ⁢ ( s ) := 1 [ - ∞ , - δ ] ∪ [ δ , ∞ ] ⁢ ( s ) ⁢ ξ ⁢ ( s ) .

Lemma 4.4.

Suppose that lim k → ∞ ⁡ λ k = 0 . Take a sequence ( R k ) k in R + such that

lim k → ∞ ⁡ R k = + ∞ .

Then we have that

(4.1) lim k → ∞ ⁡ ∥ V k * ∘ π ρ k , λ k ⁢ ( a ) ∘ V k ∘ M { R k ≤ | | } ∥ op = 0 , a ∈ C * ⁢ ( G ) .

Proof.

By (3.1), it suffices to show that, for F ∈ L 1 ⁢ ( G ) with F ^ 3 , 4 ∈ C c ⁢ ( ℝ 4 ) , we have

(4.2) F ^ 3 , 4 ⁢ ( t , ε ⁢ e t ⁢ s λ k - ε ⁢ x λ k , - 1 2 ⁢ ( ε ⁢ x + ε ⁢ e t ⁢ s ) , λ k ) = 0

for | x | ≥ R k and k large enough. Now if s , x have the same sign, we have that | ε ⁢ x + ε ⁢ e t ⁢ s | ≥ R k , and thus (4.2) is satisfied. Similarly, if s and x have different signs, it follows that

| e t ⁢ s λ k - x λ k | ≥ R k | λ k | . ∎

Lemma 4.5.

Suppose that lim k → ∞ ⁡ λ k = 0 and lim k → ∞ ⁡ ρ k ⁢ λ k = ω ∈ R * . Take a sequence ( R k ) k in R + such that

(4.3) lim k → ∞ ⁡ R k = + ∞ 𝑎𝑛𝑑 lim k → ∞ ⁡ R k ⁢ | λ k | = 0 .

Then we have that

(4.4) lim k → ∞ ⁡ ∥ π ρ k , λ k ⁢ ( a ) ∘ V k ∘ M { | | ≤ | R k λ k | } ∥ op = 0 , a ∈ C * ⁢ ( G ) .

Proof.

We prove the lemma first for F ∈ L 1 ⁢ ( G ) such that F ^ 3 , 4 ∈ C c ∞ ⁢ ( ℝ 4 ) . Let M I k := M { | | ≤ R k | λ k | } be the multiplication operator on L 2 ⁢ ( ℝ , d ⁢ x | x | ) with the characteristic function of the interval I k := [ - R k ⁢ | λ k | , R k ⁢ | λ k | ] . By applying relation (3.7) with sequences ( ρ k ) k and ( λ k ) k in ℝ , the operator π ρ k , λ k ⁢ ( F ) ∘ V k ∘ M I k is a kernel operator with kernel H k ⁢ ( s , x ) ⁢ 1 I k ⁢ ( x ) , and

| H k ⁢ ( s , x ) ⁢ 1 I k ⁢ ( x ) | ≤ 1 I k ⁢ ( x ) ⁢ 1 | ρ k ⁢ λ k | ⁢ ψ ⁢ ( λ k ) ⁢ ∫ ℝ ψ ⁢ ( t ) ⁢ ψ ⁢ ( x - e t ⁢ s λ k ) ⁢ ψ ⁢ ( x + e t ⁢ s ) ⁢ 𝑑 t

for s , x ∈ ℝ and k ∈ ℕ , where ψ is a positive continuous function with compact support on ℝ given in (3.8). Now for any C > 0 large enough, we have that

| x - e t ⁢ s λ k | ≥ R k

if t ∈ supp ⁡ ( ψ ) , | x | ≤ R k ⁢ | λ k | , and | s | ≥ C ⁢ R k ⁢ | λ k | . Therefore, it follows that

sup x ∈ I k ⁡ ∫ ℝ | H k ⁢ ( s , x ) | ⁢ 𝑑 s ≤ 1 | ρ k ⁢ λ k | ⁢ ∥ ψ ∥ ∞ 2 ⁢ sup x ∈ I k ⁡ ∫ ℝ ∫ ℝ ψ ⁢ ( t ) ⁢ ψ ⁢ ( x - e t ⁢ s λ k ) ⁢ 𝑑 t ⁢ 𝑑 s

= 1 | ρ k ⁢ λ k | ⁢ ∥ ψ ∥ ∞ 2 ⁢ sup x ∈ I k ⁡ ∫ C ⁢ I k ∫ ℝ ψ ⁢ ( t ) ⁢ ψ ⁢ ( x - e t ⁢ s λ k ) ⁢ 𝑑 t ⁢ 𝑑 s

≤ C ′ ⁢ R k ⁢ | λ k |

for a new constant C ′ > 0 (independent of k). Furthermore,

for another constant C ′′ > 0 . Hence, Young’s condition tells us that

lim k → ∞ ⁡ ∥ π ρ k , λ k ⁢ ( F ) ∘ V k ∘ M { | | ≤ | R k λ k | } ∥ op = 0 .

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 thus there exists N ε ∈ ℕ such that

∥ π ρ k , λ k ⁢ ( F ε ) ∘ V k ∘ M I k ∥ op < ε for ⁢ k > N ε .

Hence,

∥ π ρ k , λ k ⁢ ( a ) ∘ V k ∘ M I k ∥ op < 2 ⁢ ε for ⁢ k > N ε . ∎

4.3 Convergence in operator norm: ω ≠ 0

In this section, we consider sequences ( λ k ) k and ( ρ k ) k satisfying

lim k → ∞ ⁡ λ k = 0 and lim k → ∞ ⁡ ω k = lim k → ∞ ⁡ λ k ⁢ ρ k = ω ≠ 0 .

Recall the equivalent representations τ μ , ν ± given in (3.14). We have the following lemma.

Lemma 4.6.

Let ( π ρ k , λ k ) k be a properly converging sequence in G ^ such that λ k = ε ⁢ | λ k | , k ∈ N , lim k → ∞ ⁡ λ k = 0 and

lim k → ∞ ⁡ ω k := lim k → ∞ ⁡ λ k ⁢ ρ k = ω ∈ ℝ * .

Take a sequence ( R k ) k in R + such that lim k → ∞ ⁡ R k = ∞ and

(4.5) lim k → ∞ ⁡ R k ⁢ | λ k | = 0 , lim k → ∞ ⁡ ( R k 2 ⁢ | λ k | ) = ∞

or, equivalently,

lim k → ∞ ⁡ ρ k R k 2 = 0 .

Then, for any F ∈ L 1 ⁢ ( G ) with F ^ 3 , 4 ∈ C c ⁢ ( R 4 ) , we have that

∥ π ρ k , λ k ⁢ ( F ) ∘ V k ∘ M { ≥ | λ k | R k } - V k ∘ τ ε ⁢ ω k , - ε + ⁢ ( F ) ∘ M { ≥ | λ k | R k } ∥ op ≤ C ⁢ ( | ω k | | R k 2 ⁢ λ k | + R k - 1 / 2 ) ,

∥ π ρ k , λ k ⁢ ( F ) ∘ V k ∘ M { ≤ - | λ k | R k } - V k ∘ τ - ε ⁢ ω k , ε - ⁢ ( F ) ∘ M { ≤ - | λ k | R k } ∥ op ≤ C ⁢ ( | ω k | | R k 2 ⁢ λ k | + R k - 1 / 2 )

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 | + | s | + | x | + | λ | ≥ D

for some D > 0 . Take C 1 > 0 such that e D ⁢ C 1 < 1 .

Let s , t , x ∈ ℝ be such that | t | ≤ D , | x | ≤ D and | s | ≥ C 1 ⁢ | R k ⁢ λ k | . We have that

(4.6)

ln ⁡ ( | e t ⁢ s - λ k ⁢ x | ) - ln ⁡ ( | e t ⁢ s | ) + λ k ⁢ x ⁢ ( e t ⁢ s ) - 1 = - λ k ⁢ x ⁢ ∫ 0 1 ( 1 e t ⁢ s - λ k ⁢ x ⁢ v - 1 e t ⁢ s ) ⁢ 𝑑 v

= - ( λ k ⁢ x ) 2 e t ⁢ s ⁢ ∫ 0 1 v ⁢ d ⁢ v e t ⁢ s - λ k ⁢ x ⁢ v

= : λ k c k ( t , s , x ) ,

where

c k ⁢ ( t , s , x ) = - λ k ⁢ x 2 e t ⁢ s ⁢ ∫ 0 1 v ⁢ d ⁢ v e t ⁢ s - λ k ⁢ x ⁢ v .

Hence for k large enough and for some new constant C = D 2 2 ⁢ C 1 2 , we have

| ln ⁡ ( | e t ⁢ s - λ k ⁢ x | ) - ln ⁡ ( | e t ⁢ s | ) + λ k ⁢ x ⁢ | e t ⁢ s | - 1 | = | λ k ⁢ c k ⁢ ( t , s , x ) | ≤ C ⁢ λ k 2 s 2 ≤ C ⁢ 1 R k 2

for all our s , x , t in ℝ , whence

(4.7) | e i ⁢ ρ k ⁢ λ k ⁢ c k ⁢ ( t , s , x ) - 1 | ≤ | ρ k ⁢ λ k ⁢ c k ⁢ ( t , s , x ) | ≤ C ⁢ | ρ k ⁢ λ k | | λ k ⁢ R k 2 | .

By (3.7), we have that

V k * ⁢ ( π ρ k , λ k ⁢ ( F ) ⁢ ( V k ⁢ ( η ) ) ) ⁢ ( s )

= e - i ⁢ ρ k ⁢ ln ⁡ ( | s | ) ⁢ | s | 1 / 2 | λ k | ⁢ ∫ ℝ ∫ ℝ e - i ⁢ ρ k ⁢ t ⁢ e t / 2 ⁢ F ^ 3 , 4 ⁢ ( t , - ε λ k ⁢ ( x - e t ⁢ s ) , - ε 2 ⁢ ( x + e t ⁢ s ) , λ k ) ⁢ 𝑑 t ⁢ e i ⁢ ρ k ⁢ ln ⁡ ( | x | ) | x | 1 / 2 ⁢ η ⁢ ( x ) ⁢ 𝑑 x

for η ∈ L 2 ⁢ ( ℝ , d ⁢ x | x | ) . It follows for our constant C 1 that, for any

η ∈ L 2 ⁢ ( ℝ , d ⁢ x | x | ) with ⁢ η ⁢ ( x ) = 0 ,

| x | ≤ R k ⁢ | λ k | and k large enough,

F ^ 3 , 4 ⁢ ( t , - ε λ k ⁢ ( x - e t ⁢ s ) , - ε 2 ⁢ ( x + e t ⁢ s ) , λ k ) ⁢ η ⁢ ( x ) = 0

for every t , x , λ k ∈ ℝ and | s | < C 1 ⁢ R k ⁢ | λ k | . For | s | ≥ C 1 ⁢ R k ⁢ | λ k | , we have that

| V k * ⁢ ( π ρ k , λ k ⁢ ( F ) ⁢ ( V k ⁢ ( η ) ) ) ⁢ ( s ) - ∫ ℝ ∫ ℝ F ^ 3 , 4 ⁢ ( t , x , λ k 2 ⁢ x - ε ⁢ e t ⁢ s , λ k ) ⁢ | e t ⁢ s | 1 / 2 ⁢ e - i ⁢ ε ⁢ ρ k ⁢ λ k ⁢ x ⁢ ( e t ⁢ s ) - 1 | - ε ⁢ λ k ⁢ x + e t ⁢ s | 1 / 2 ⁢ η ⁢ ( - ε ⁢ λ k ⁢ x + e t ⁢ s ) ⁢ 𝑑 x ⁢ 𝑑 t |

= | ∫ ℝ ∫ ℝ F ^ 3 , 4 ( t , x , λ k 2 x - ε e t s , λ k ) | e t s | 1 / 2

⋅ ( e - i ⁢ ε ⁢ ρ k ⁢ λ k ⁢ x ⁢ ( | e t ⁢ s | ) - 1 + i ⁢ ε ⁢ ρ k ⁢ λ k ⁢ c k ⁢ ( t , s , ε ⁢ x ) - e - i ⁢ ε ⁢ ρ k ⁢ λ k ⁢ x ⁢ ( | e t ⁢ s | ) - 1 | - ε ⁢ λ k ⁢ x + e t ⁢ s | 1 / 2 ) η ( - ε λ k x + e t s ) d x d t |

≤ ∫ ℝ ∫ ℝ | F ^ 3 , 4 ⁢ ( t , x , λ k 2 ⁢ x - ε ⁢ e t ⁢ s , λ k ) | ⁢ | e i ⁢ ε ⁢ ρ k ⁢ λ k ⁢ c k ⁢ ( t , s , ε ⁢ x ) - 1 | | - ε ⁢ λ k ⁢ x + e t ⁢ s | 1 / 2 ⁢ | e t ⁢ s | 1 / 2 ⁢ | η ⁢ ( - ε ⁢ λ k ⁢ x + e t ⁢ s ) | ⁢ 𝑑 x ⁢ 𝑑 t

≤ C ⁢ | ρ k ⁢ λ k | | λ k ⁢ R k 2 | ⁢ ∫ ℝ ∫ ℝ | F ^ 3 , 4 ⁢ ( t , x , λ k 2 ⁢ x - ε ⁢ e t ⁢ s , λ k ) | ⁢ 1 | - ε ⁢ λ k ⁢ x + e t ⁢ s | 1 / 2 ⁢ | e t ⁢ s | 1 / 2 ⁢ | η ⁢ ( - ε ⁢ λ k ⁢ x + e t ⁢ s ) | ⁢ 𝑑 x ⁢ 𝑑 t ,

where the last inequality follows by (4.7). Therefore, for any

η ∈ L 2 ⁢ ( ℝ , d ⁢ x | x | ) , with ⁢ η ⁢ ( x ) = 0 ,

for | x | ≤ R k ⁢ | λ k | and k large enough, we deduce that

∫ { | s | ≥ C 1 R k | λ k | } | V k * ( π ρ k , λ k ( F ) ( V k ( η ) ) ) ( s )

- ∫ ℝ ∫ ℝ F ^ 3 , 4 ( t , x , λ k 2 x - ε e t s , λ k ) | e t ⁢ s | 1 / 2 ⁢ e - i ⁢ ε ⁢ ρ k ⁢ λ k ⁢ x ⁢ ( e t ⁢ s ) - 1 | - ε ⁢ λ k ⁢ x + e t ⁢ s | 1 / 2 η ( - ε λ k x + e t s ) d x d t | 2 d ⁢ s | s |

≤ ∫ { | s | ≥ C 1 R k | λ k | } ( C ⁢ | ρ k ⁢ λ k | | λ k ⁢ R k 2 | ⁢ ∫ ℝ ∫ ℝ | F ^ 3 , 4 ⁢ ( t , x , λ k 2 ⁢ x - ε ⁢ e t ⁢ s , λ k ) | ⁢ 1 | - ε ⁢ λ k ⁢ x + e t ⁢ s | 1 / 2 ⁢ | e t ⁢ s | 1 / 2 ⁢ | η ⁢ ( - ε ⁢ λ k ⁢ x + e t ⁢ s ) | ⁢ 𝑑 x ⁢ 𝑑 t ) 2 ⁢ d ⁢ s | s |

≤ ( C ⁢ | ρ k ⁢ λ k | | λ k ⁢ R k 2 | ) 2 ⁢ ∫ { | s | ≥ C 1 R k | λ k | } ∫ ℝ ∫ ℝ | e t / 2 ⁢ F ^ 3 , 4 ⁢ ( t , x , λ k 2 ⁢ x - ε ⁢ e t ⁢ s , λ k ) | ⁢ 1 | - ε ⁢ λ k ⁢ x + e t ⁢ s | ⁢ | η ⁢ ( - ε ⁢ λ k ⁢ x + e t ⁢ s ) | 2 ⁢ 𝑑 x ⁢ 𝑑 t ⁢ 𝑑 s

⋅ sup | s | ≥ C 1 ⁢ R k ⁢ | λ k | ∫ ℝ ∫ ℝ e r / 2 | F ^ 3 , 4 ( r , u , λ k u - ε e r s , λ k ) | d u d r .

Choose a positive real-valued function φ ∈ C c ∞ ⁢ ( ℝ , ℝ + ) such that

| e t / 2 ⁢ F ^ 3 , 4 ⁢ ( t , x , u , λ k ) | ≤ φ ⁢ ( t ) ⁢ φ ⁢ ( x ) ⁢ φ ⁢ ( u ) for ⁢ k ∈ ℕ ⁢ and ⁢ u , t , x ∈ ℝ .

Then we obtain the inequality

(4.8)

∫ { | s | ≥ C 1 R k | λ k | } | V k * ( π ρ k , λ k ( F ) ( V k ( η ) ) ) ( s )

≤ ( C ⁢ | ρ k ⁢ λ k | | λ k | ⁢ R k 2 ) 2 ⁢ ∫ ℝ ∫ ℝ ∫ ℝ e - t ⁢ φ ⁢ ( t ) ⁢ φ ⁢ ( x ) ⁢ 1 | s | ⁢ | η ⁢ ( e t ⁢ s ) | 2 ⁢ 𝑑 x ⁢ 𝑑 t ⁢ 𝑑 s

≤ ( C ⁢ | ρ k ⁢ λ k | | λ k ⁢ R k 2 | ) 2 ⁢ ∥ η ∥ 2 2

for some new constant C > 0 depending on F.

Now

e t ⁢ s - ε ⁢ λ k ⁢ x = e t ⁢ s ⁢ α k ⁢ ( t , s , x ) = e t + ln ⁡ ( α k ⁢ ( t , s , x ) ) ⁢ s ,

where

α k ⁢ ( t , s , x ) = 1 - ε ⁢ λ k ⁢ x e t ⁢ s .

If | s | ≥ C 1 ⁢ | λ k | ⁢ R k for the above constant C 1 , k large enough and for all | t | ≤ D and | x | ≤ D , it follows that

| λ k ⁢ x e t ⁢ s | ≤ e D ⁢ | λ k | ⁢ D C 1 ⁢ | λ k | ⁢ R k = D C 1 ⁢ R k ,

(4.9) | ln ⁡ ( α k ⁢ ( t , s , x ) ) | ≤ C ⁢ 1 R k ,

| ∂ t ⁡ ln ⁡ ( α k ⁢ ( t , s , x ) ) | ≤ C ⁢ 1 R k

for some (new) constant C > 1 .

Let

r = τ k ⁢ ( t ) := t + ln ⁡ ( α k ⁢ ( t , s , x ) ) ,

τ k - 1 ⁢ ( r ) = μ k ⁢ ( r ) = t for ⁢ | t | ≤ D , | x | ≤ D , | s | ≥ C 1 ⁢ | λ k ⁢ R k | .

We have that

| μ k ⁢ ( r ) - r | = | ln ⁡ ( α k ⁢ ( t , s , x ) ) | ≤ C ⁢ 1 R k .

Then

∫ ℝ ∫ ℝ F ^ 3 , 4 ⁢ ( t , x , λ k 2 ⁢ x - ε ⁢ e t ⁢ s , λ k ) ⁢ | e t ⁢ s | 1 / 2 ⁢ e - i ⁢ ε ⁢ ρ k ⁢ λ k ⁢ x ⁢ ( e t ⁢ s ) - 1 | - ε ⁢ λ k ⁢ x + e t ⁢ s | 1 / 2 ⁢ η ⁢ ( - ε ⁢ λ k ⁢ x + e t ⁢ s ) ⁢ 𝑑 x ⁢ 𝑑 t

= ∫ ℝ ∫ ℝ F ^ 3 , 4 ⁢ ( μ k ⁢ ( r ) , x , e r ⁢ s , λ k ) ⁢ | e μ k ⁢ ( r ) ⁢ s | 1 / 2 ⁢ e - i ⁢ ε ⁢ ρ k ⁢ λ k ⁢ x ⁢ ( e μ k ⁢ ( r ) ⁢ s ) - 1 | e r ⁢ s | 1 / 2 ⁢ η ⁢ ( e r ⁢ s ) ⁢ 𝑑 x ⁢ μ k ′ ⁢ ( r ) ⁢ 𝑑 r .

Now by (4.9), for k large enough, we have

| F ^ 3 , 4 ⁢ ( μ k ⁢ ( r ) , x , e r ⁢ s , λ k ) ⁢ | e μ k ⁢ ( r ) ⁢ s | 1 / 2 ⁢ e - i ⁢ ε ⁢ ρ k ⁢ λ k ⁢ x ⁢ ( e μ k ⁢ ( r ) ⁢ s ) - 1 ⁢ μ k ′ ⁢ ( r ) - F ^ 3 , 4 ⁢ ( r , x , e r ⁢ s , 0 ) ⁢ | e r ⁢ s | 1 / 2 ⁢ e - i ⁢ ε ⁢ ρ k ⁢ λ k ⁢ x ⁢ ( e r ⁢ s ) - 1 |

≤ C R k ⁢ φ ⁢ ( r ) ⁢ φ ⁢ ( x ) ⁢ | s | 1 / 2 ,

where φ ∈ C c ∞ ⁢ ( ℝ , ℝ + ) is given before (4.8). Therefore, for k large enough and any η ∈ L 2 ⁢ ( ℝ , d ⁢ s | s | ) with ∥ η ∥ 2 ≤ 1 and η ⁢ ( u ) = 0 for | u | ≤ R k ⁢ | λ k | , we have that

- ∫ ℝ ∫ ℝ F ^ 3 , 4 ( r , x , - e r s , 0 ) e - i ⁢ ε ⁢ ρ k ⁢ λ k ⁢ x ⁢ ( e r ⁢ s ) - 1 η ( e r s ) d x d r | 2 d ⁢ s | s |

- ∫ ℝ ∫ ℝ F ^ 3 , 4 ( r , x , - e r s , 0 ) e - i ⁢ ε ⁢ ρ k ⁢ λ k ⁢ x ⁢ ( e r ⁢ s ) - 1 η ( e r s ) d x d r | 2 d ⁢ s | s |

(4.10) ≤ C R k

for a new constant C > 0 depending on F.

Let us recall that, for η ∈ L 2 ⁢ ( ℝ σ , d ⁢ x | x | ) (see (3.14)),

τ μ , ν σ ⁢ ( F ) ⁢ η ⁢ ( u ) = ∫ ℝ F ^ 2 , 3 , 4 ⁢ ( t , μ ⁢ σ ⁢ ( e t ⁢ u ) - 1 , σ ⁢ ν ⁢ ( e t ⁢ u ) , 0 ) ⁢ η ⁢ ( e t ⁢ u ) ⁢ 𝑑 t , u ∈ ℝ σ .

Hence for any η ∈ L 2 ⁢ ( ℝ + , d ⁢ s | s | ) , we have that

∥ V k * ∘ π ρ k , λ k ⁢ ( F ) ∘ V k ∘ M { ≥ | R k λ k | } ⁢ ( η ) - τ ε ⁢ ω k , - ε + ⁢ ( F ) ∘ M { ≥ | λ k | R k } ⁢ ( η ) ∥ 2

= ( ∫ ℝ | V k * ∘ π ρ k , λ k ⁢ ( F ) ∘ V k ∘ M { ≥ | R k λ k | } ⁢ ( η ) ⁢ ( s ) - ∫ ℝ F ^ 2 , 3 , 4 ⁢ ( t , ε ⁢ ω k ⁢ s - 1 ⁢ e - t , - s ⁢ e t , 0 ) ⁢ M { ≥ | λ k | R k } ⁢ ( η ) ⁢ ( e t ⁢ s ) ⁢ 𝑑 t | 2 ⁢ d ⁢ s | s | ) 1 / 2

= ( ∫ ℝ | V k * ∘ π ρ k , λ k ( F ) ∘ V k ∘ M { ≥ | R k λ k | } ( η ) ( s )

- ∫ ℝ ∫ ℝ F ^ 3 , 4 ( t , x , - e t s , 0 ) | e t s | 1 / 2 e - i ⁢ ε ⁢ ρ k ⁢ λ k ⁢ x ⁢ ( e t ⁢ s ) - 1 M { ≥ | λ k | R k } ( η ) ( e t s ) d t d x | 2 d ⁢ s | s | ) 1 / 2

≤ ( C R k ) 1 / 2 ∥ η ∥ 2 + ( ∫ ℝ | V k * ∘ π ρ k , λ k ( F ) ∘ V k ∘ M { ≥ | R k λ k | } ( η ) ( s ) - ∫ ℝ ∫ ℝ F ^ 3 , 4 ( t , x , λ k 2 x - ε e t s , 0 )

⋅ | e t ⁢ s | 1 / 2 ⁢ e - i ⁢ ε ⁢ ρ k ⁢ λ k ⁢ x ⁢ ( e t ⁢ s ) - 1 | - ε ⁢ λ k ⁢ x + e t ⁢ s | 1 / 2 M { ≥ R k | λ k | } ( η ) ( - ε λ k x + e t s ) d x d t | 2 d ⁢ s | s | ) 1 / 2 (by (4.10))

≤ ( C R k ) 1 / 2 ∥ η ∥ 2 + ( ∫ ℝ | ∫ ℝ ∫ ℝ F ^ 3 , 4 ( t , x , λ k 2 x - ε e t s , 0 )

⋅ | e t ⁢ s | 1 / 2 ⁢ e - i ⁢ ε ⁢ ρ k ⁢ λ k ⁢ x ⁢ ( e t ⁢ s ) - 1 + i ⁢ ρ k ⁢ λ k ⁢ c k ⁢ ( t , s , ε ⁢ x ) | - ε ⁢ λ k ⁢ x + e t ⁢ s | 1 / 2 ⁢ M { ≥ | R k λ k | } ⁢ η ⁢ ( - ε ⁢ λ k ⁢ x + e t ⁢ s ) ⁢ d ⁢ x ⁢ d ⁢ t

- ∫ ℝ ∫ ℝ F ^ 3 , 4 ( t , x , λ k 2 x - ε e t s , 0 ) | e t ⁢ s | 1 / 2 ⁢ e - i ⁢ ε ⁢ ρ k ⁢ λ k ⁢ x ⁢ ( e t ⁢ s ) - 1 | - ε ⁢ λ k ⁢ x + e t ⁢ s | 1 / 2 M { ≥ R k | λ k | } η ( - ε λ k x + e t s ) d x d t | 2 d ⁢ s | s | ) 1 / 2

≤ ( C R k ) 1 / 2 ∥ η ∥ 2 + ( ∫ ℝ | ∫ ℝ ∫ ℝ F ^ 3 , 4 ( t , x , λ k 2 x - ε e t s , 0 )

⋅ | e i ⁢ ε ⁢ ρ k ⁢ λ k ⁢ c k ⁢ ( t , s , ε ⁢ x ) - 1 | | e t ⁢ s | 1 / 2 ⁢ e - i ⁢ ε ⁢ ρ k ⁢ λ k ⁢ x ⁢ ( e t ⁢ s ) - 1 | - ε ⁢ λ k ⁢ x + e t ⁢ s | 1 / 2 M { ≥ | R k λ k | } η ( - ε λ k x + e t s ) d x d t | 2 d ⁢ s | s | ) 1 / 2

≤ C ⁢ ( | ρ k ⁢ λ k | | λ k ⁢ R k 2 | + R k - 1 / 2 ) ⁢ ∥ η ∥ 2 (by (4.7))

for a constant C > 0 . Then, for any s ∈ ℝ , it follows that

C ⁢ ( | ω k | | R k 2 ⁢ λ k | + R k - 1 / 2 ) ≥ ∥ π ρ k , λ k α ⁢ ( F ) ∘ V k ∘ M { ≥ | λ k | R k } - V k ∘ ( τ ε ⁢ ω k , - ε + ) α ⁢ ( F ) ∘ M { ≥ | λ k | R k } ∥ op

= ∥ S ∘ π ρ k , λ k ⁢ ( F ) ∘ S ∘ V k ∘ M { ≥ | λ k | R k } - V k ∘ τ - ε ⁢ ω k , ε + ⁢ ( F ) ∘ M { ≥ | λ k | R k } ∥ op (by (3.4), (3.12))

= ∥ π ρ k , λ k ⁢ ( F ) ∘ V k ∘ M { ≤ - | λ k | R k } - V k ∘ τ - ε ⁢ ω k , ε - ⁢ ( F ) ∘ M { ≤ - | λ k | R k } ∥ op (by (3.5), (3.13)) .

This concludes the proof. ∎

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

L = { O - ω , 1 , 0 , O ω , - 1 , 0 } .

For k ∈ ℕ , let λ k = ε ⁢ | λ k | and let ( R k ) k be a sequence in ℝ + satisfying the conditions given in (4.5), namely

lim k → ∞ ⁡ R k ⁢ | λ k | = 0 and lim k → ∞ ⁡ ( R k 2 ⁢ | λ k | ) = ∞ .

We define σ k ω by

(4.11) σ k ω ( ϕ | L ) := V k ∘ ( ϕ ( τ ε ⁢ ω k , - ε + ) ∘ M { ≥ | λ k | R k } ⊕ ϕ ( τ - ε ⁢ ω k , ε - ) ∘ M { ≤ - | λ k | R k } ) ∘ V k * , ϕ ∈ l ∞ ( G ^ ) ,

where τ μ , ν σ acting on L 2 ⁢ ( ℝ σ , d ⁢ u | u | ) is given in (3.11).

Theorem 4.7.

Suppose that lim k → ∞ ⁡ ρ k ⁢ λ k = ω ∈ R * and λ k = ε ⁢ | λ k | for k ∈ N . Then

lim k → ∞ ⁡ ∥ π ρ k , λ k ⁢ ( a ) - σ k ω ⁢ ( a ^ | L ) ∥ op = 0

for every a ∈ C * ⁢ ( G ) .

Proof.

Apply Proposition 4.1 and Lemma 4.6. Let a ∈ C * ⁢ ( G ) and ε > 0 . Choose F ∈ L 1 ⁢ ( G ) with F ^ 3 , 4 ∈ C c ⁢ ( ℝ 4 ) such that ∥ a - F ∥ C * ⁢ ( G ) < ε . Then, by Lemma 4.6, there exists an N ε ∈ ℕ such that

∥ π ρ k , λ k ⁢ ( F ) - σ k ω ⁢ ( F ^ | L ) ∥ op < ε for ⁢ k ≥ N ε .

Hence,

∥ π ρ k , λ k ⁢ ( a ) - σ k ω ⁢ ( a ^ | L ) ∥ op < 2 ⁢ ε for ⁢ k ≥ N ε . ∎

4.4 Convergence in operator norm: ω = 0

Suppose now that

lim k → ∞ ⁡ λ k = 0 = lim k → ∞ ⁡ λ k ⁢ ρ k .

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 ( R k ) k in ℝ + .

Lemma 4.8.

Suppose that lim k → ∞ ⁡ λ k = 0 . Take a sequence ( R k ) k in R + such that

(4.12) lim k → ∞ ⁡ R k = + ∞ , lim k → ∞ ⁡ R k 2 ⁢ | λ k | = 0 .

Then we have that

(4.13) lim k → ∞ ⁡ ∥ π ρ k , λ k ⁢ ( a ) ∘ V k ∘ M { | | ≤ | R k λ k | } ∥ op = 0 , a ∈ C * ⁢ ( G ) .

Proof.

It suffices to consider F ∈ L 1 ⁢ ( G ) such that F ^ 3 , 4 ∈ C c ∞ ⁢ ( ℝ 4 ) . There exists M > 0 such that

F ^ 3 , 4 ⁢ ( t , x , y , λ ) = 0

whenever | t | + | x | + | y | + | λ | ≥ M . We have that

(4.14)

V k * ⁢ ( π ρ k , λ k ⁢ ( F ) ⁢ ( V k ⁢ ( η ) ) ) ⁢ ( s ) = e - i ⁢ ρ k ⁢ ln ⁡ ( | s | ) ⁢ | s | 1 / 2 | λ k | 1 / 2 ⁢ ∫ ℝ ( ∫ ℝ e t / 2 ⁢ e - i ⁢ ρ k ⁢ t ⁢ F ^ 3 , 4 ⁢ ( t , ε ⁢ e t ⁢ s λ k - x , - λ k 2 ⁢ ( x + ε ⁢ e t ⁢ s λ k ) , λ k ) ⁢ 𝑑 t )

⋅ e i ⁢ ρ k ⁢ ln ⁡ ( | λ k ⁢ x | ) | x | 1 / 2 ⁢ η ⁢ ( ε ⁢ λ k ⁢ x ) ⁢ d ⁢ x

= e - i ⁢ ρ k ⁢ ln ⁡ ( | s | ) ⁢ | s | 1 / 2 | λ k | 1 / 2 ⁢ ∫ ℝ ( ∫ ℝ e t / 2 ⁢ e - i ⁢ ρ k ⁢ t ⁢ F ^ 3 , 4 ⁢ ( t , ε ⁢ e t ⁢ s λ k - ε ⁢ x λ k , - 1 2 ⁢ ( ε ⁢ x + ε ⁢ e t ⁢ s ) , λ k ) ⁢ 𝑑 t )

⋅ e i ⁢ ρ k ⁢ ln ⁡ ( | x | ) ⁢ | x | 1 / 2 ⁢ η ⁢ ( x ) ⁢ d ⁢ x | x | .

Hence for k large enough, | t | ≤ M , | x | ≤ R k ⁢ | λ k | , and | s | ≥ C ⁢ R k ⁢ | λ k | for some constant C > 3 ⁢ e M , we have that

| ε ⁢ e t ⁢ s λ k - ε ⁢ x λ k | > M ,

and so

F ^ 3 , 4 ⁢ ( t , ε ⁢ e t ⁢ s λ k - ε ⁢ x λ k , - 1 2 ⁢ ( ε ⁢ x + ε ⁢ e t ⁢ s ) , λ k ) = 0 .

Thus,

M { | | ≥ C R k | λ k | } ∘ V k * ∘ π ρ k , λ k ⁢ ( F ) ∘ V k ∘ M { | | ≤ R k | λ k | } = 0

for k large enough. Choose an even function φ : ℝ → ℝ + in C c ⁢ ( ℝ ) with compact support such that

| F ^ 3 , 4 ⁢ ( t , x , y , λ ) | ≤ φ ⁢ ( t ) ⁢ φ ⁢ ( x ) ⁢ φ ⁢ ( y ) , t , x , y , λ ∈ ℝ .

Now, by Young’s inequality,

∥ M { | | ≤ C R k | λ k | } ∘ V k * ∘ π ρ k , λ k ⁢ ( F ) ∘ V k ∘ M { | | ≤ R k | λ k | } ∥ op

≤ sup | x | ≤ R k ⁢ | λ k | ⁡ ∫ | s | ≤ C ⁢ R k ⁢ | λ k | | x ⁢ s | 1 / 2 | λ k | 1 / 2 ⁢ ( ∫ ℝ e t / 2 ⁢ | F ^ 3 , 4 ⁢ ( t , ε ⁢ e t ⁢ s λ k - ε ⁢ x λ k , - 1 2 ⁢ ( ε ⁢ x + ε ⁢ e t ⁢ s ) , λ k ) | ⁢ 𝑑 t ) ⁢ d ⁢ s | s |

+ sup | s | ≤ C ⁢ R k ⁢ | λ k | ⁡ ∫ | x | ≤ R k ⁢ | λ k | | x ⁢ s | 1 / 2 | λ k | 1 / 2 ⁢ ( ∫ ℝ e t / 2 ⁢ | F ^ 3 , 4 ⁢ ( t , ε ⁢ e t ⁢ s λ k - ε ⁢ x λ k , - 1 2 ⁢ ( ε ⁢ x + ε ⁢ e t ⁢ s ) , λ k ) | ⁢ 𝑑 t ) ⁢ d ⁢ x | x |

≤ sup | x | ≤ R k ⁢ | λ k | ⁡ ∫ | s | ≤ C ⁢ R k ⁢ | λ k | | x | 1 / 2 | λ k ⁢ s | 1 / 2 ⁢ ( ∫ ℝ e t / 2 ⁢ φ ⁢ ( t ) ⁢ φ ⁢ ( e t ⁢ s λ k - x λ k ) ⁢ φ ⁢ ( 1 2 ⁢ ( x + e t ⁢ s ) ) ⁢ 𝑑 t ) ⁢ 𝑑 s

+ sup | s | ≤ C ⁢ R k ⁢ | λ k | ⁡ ∫ | x | ≤ R k ⁢ | λ k | | s | 1 / 2 | λ k ⁢ x | 1 / 2 ⁢ ( ∫ ℝ e t / 2 ⁢ φ ⁢ ( t ) ⁢ φ ⁢ ( e t ⁢ s λ k - x λ k ) ⁢ φ ⁢ ( 1 2 ⁢ ( x + e t ⁢ s ) ) ⁢ 𝑑 t ) ⁢ 𝑑 x

≤ R k 1 / 2 ⁢ ∫ | s | ≤ C ⁢ R k ⁢ | λ k | 1 | s | 1 / 2 ⁢ ( ∫ ℝ e t / 2 ⁢ φ ⁢ ( t ) ⁢ φ ⁢ ( e t ⁢ s λ k - x λ k ) ⁢ φ ⁢ ( 1 2 ⁢ ( x + e t ⁢ s ) ) ⁢ 𝑑 t ) ⁢ 𝑑 s

+ C 1 / 2 ⁢ R k 1 / 2 ⁢ ∫ | x | ≤ R k ⁢ | λ k | 1 | x | 1 / 2 ⁢ ( ∫ ℝ e t / 2 ⁢ φ ⁢ ( t ) ⁢ φ ⁢ ( e t ⁢ s λ k - x λ k ) ⁢ φ ⁢ ( 1 2 ⁢ ( x + e t ⁢ s ) ) ⁢ 𝑑 t ) ⁢ 𝑑 x

≤ R k 1 / 2 ⁢ ∥ φ ∥ ∞ 2 ⁢ ∫ | s | ≤ C ⁢ R k ⁢ | λ k | 1 | s | 1 / 2 ⁢ 𝑑 s ⁢ ∫ ℝ e t / 2 ⁢ φ ⁢ ( t ) ⁢ 𝑑 t + C 1 / 2 ⁢ R k 1 / 2 ⁢ ∥ φ ∥ ∞ 2 ⁢ ∫ | x | ≤ R k ⁢ | λ k | 1 | x | 1 / 2 ⁢ 𝑑 x ⁢ ∫ ℝ e t / 2 ⁢ φ ⁢ ( t ) ⁢ 𝑑 t

= R k 1 / 2 ⁢ ∥ φ ∥ ∞ 2 ⁢ 2 ⁢ C 1 / 2 ⁢ R k 1 / 2 ⁢ | λ k | 1 / 2 ⁢ ∫ ℝ e t / 2 ⁢ φ ⁢ ( t ) ⁢ 𝑑 t + C 1 / 2 ⁢ R k 1 / 2 ⁢ ∥ φ ∥ ∞ 2 ⁢ 2 ⁢ R k 1 / 2 ⁢ | λ k | 1 / 2 ⁢ ∫ ℝ e t / 2 ⁢ φ ⁢ ( t ) ⁢ 𝑑 t

≤ C ′ ⁢ R k ⁢ | λ k | 1 / 2

for some constant C ′ > 0 . ∎

Lemma 4.9.

Let ( ρ k ) k be a real sequence with lim k → ∞ ⁡ ρ k = 0 . Then, for any real sequence ( λ k ) k , we have that

lim k → ∞ ⁡ ∥ π ρ k , λ k ⁢ ( a ) - π 0 , λ k ⁢ ( a ) ∥ op = 0 , a ∈ C * ⁢ ( G ) .

Proof.

For k ∈ ℕ , the identity

π ρ k , λ k ⁢ ( F ) - π 0 , λ k ⁢ ( F ) = ∫ G ( χ ρ k ⁢ ( g ) - 1 ) ⁢ π 0 , λ k ⁢ ( g ) ⁢ F ⁢ ( g ) ⁢ 𝑑 g , F ∈ C c ⁢ ( G ) ,

where χ ρ k ⁢ ( exp ⁡ t ⁢ T ⋅ h ) = e - i ⁢ ρ k ⁢ t for t ∈ ℝ and h ∈ H , the Heisenberg group, shows that

∥ π ρ k , λ k ⁢ ( F ) - π 0 , λ k ⁢ ( F ) ∥ op ≤ | ρ k | ⁢ ∥ F 1 ∥ 1 ,

where F 1 ⁢ ( t , x , y , z ) := t ⁢ F ⁢ ( 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 let ω k := λ k ⁢ ρ k be such that lim k → ∞ ⁡ ω k = 0 . Take a sequence ( R k ) k in R + such that lim k → ∞ ⁡ R k = ∞ and

(4.15) lim k → ∞ ⁡ R k 2 ⁢ λ k = 0 , lim k → ∞ ⁡ ω k R k 2 ⁢ | λ k | = 0 .

Let F ∈ L 1 ⁢ ( G ) be such that F ^ 3 , 4 ∈ C c ⁢ ( R 4 ) . For ε ∈ { + 1 , - 1 } , λ k = ε ⁢ | λ k | , we have that

(4.16) ∥ π ρ k , λ k ⁢ ( F ) ∘ V k ∘ M { ≥ | λ k | R k } - V k ∘ τ ε ⁢ ω k , - ε + ⁢ ( F ) ∘ M { ≥ | λ k | R k } ∥ op ≤ C ⁢ ( | ω k | | R k 2 ⁢ λ k | + R k - 1 / 2 ) ,

(4.17) ∥ π ρ k , λ k ⁢ ( F ) ∘ V k ∘ M { ≤ - | λ k | R k } - V k ∘ τ - ε ⁢ ω k , ε - ⁢ ( F ) ∘ M { ≤ - | λ k | R k } ∥ op ≤ C ⁢ ( | ω k | | R k 2 ⁢ λ k | + R k - 1 / 2 )

for some constant C depending on F.

Proof.

Inequalities (4.16) and (4.17) are proved in the same way as the corresponding ones in Lemma 4.6. ∎

Remark 4.11.

We can take, for example,

R k = 1 | λ k | 1 / 3 , k ∈ ℕ ,

if the sequence ( ρ k ) k is bounded. Otherwise, for any k ∈ ℕ , let R k 2 = | ρ k | ⁢ M k , for some M k ∈ ℝ such that lim k → ∞ ⁡ M k = ∞ and lim k → ∞ ⁡ M k ⁢ ω k = 0 . In the second case, we have that

lim k → ∞ ⁡ ω k R k 2 ⁢ λ k = lim k → ∞ ⁡ ρ k R k 2 = lim k → ∞ ⁡ 1 M k = 0 .

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 , d ⁢ μ ) with any Borel measure d ⁢ μ on ℝ .

Definition 4.12.

Suppose that lim k → ∞ ⁡ λ k = 0 , ρ k ≠ 0 , k ∈ ℕ , and lim k → ∞ ⁡ | ρ k ⁢ λ k | = 0 . As before, let ω k := λ k ⁢ ρ k for all k ∈ ℕ . Choose two sequences ( Q k ) k , ( P k ) k in ℝ + such that

lim k → ∞ ⁡ Q k = ∞ = lim k → ∞ ⁡ P k , lim k → ∞ ⁡ Q k P k = 0 , lim k → ∞ ⁡ ω k ⁢ P k = 0 , R k ≤ | ρ k | ⁢ Q k

for all k ∈ ℕ , where the sequence ( R k ) k is given as in (4.15), i.e.

lim k → ∞ ⁡ R k = ∞ , lim k → ∞ ⁡ R k 2 ⁢ λ k = 0 , lim k → ∞ ⁡ ω k R k 2 ⁢ | λ k | = 0 .

Define the subsets J k ± and I k , j ± , for k ∈ ℕ and j ∈ { 1 , 2 , 3 } , in ℝ by

J k + := ] R k | λ k | , | ω k | Q k ] , J k - := [ - | ω k | Q k , - R k | λ k | [ ,

I k , 1 + := ] 0 , | ω k | Q k ] , I k , 1 - := [ - | ω k | Q k , 0 [ ,

I k , 2 + := ] | ω k | Q k , | ω k | P k ] , I k , 2 - := [ - | ω k | P k , - | ω k | Q k [ ,

I k , 3 + := ] | ω k | P k , ∞ [ , I k , 3 - := ] - ∞ , - | ω k | P k [ .

Lemma 4.13.

Suppose that λ k = ε ⁢ | λ k | , ρ k ≠ 0 , for k ∈ N , lim k → ∞ ⁡ λ k = 0 , and lim k → ∞ ⁡ | ρ k ⁢ λ k | = 0 . Take a sequence ( R k ) k in R + and sequences ( Q k ) k , ( P k ) k ⊂ R + satisfying the conditions in Definition 4.12. Then, for any a ∈ C * ⁢ ( G ) , we have that

lim k → ∞ ⁡ ∥ τ ω k , - ε + ⁢ ( a ) - τ ω k , 0 + ⁢ ( a ) ∘ M J k + ⊕ τ 0 , 0 + ⁢ ( a ) ∘ M I k , 2 + ⊕ τ 0 , - ε ⁢ | ω k | + ⁢ ( a ) ∘ M I k , 3 + ∥ op = 0 , lim k → ∞ ⁡ ∥ τ - ω k , ε - ⁢ ( a ) - τ - ω k , 0 - ⁢ ( a ) ∘ M J k - ⊕ τ 0 , 0 - ⁢ ( a ) ∘ M I k , 2 - ⊕ τ 0 , ε ⁢ | ω k | - ⁢ ( a ) ∘ M I k , 3 - ∥ op = 0 ,

where the representation τ μ , ν ± is defined in (3.14).

Proof.

First, let F ∈ L 1 ⁢ ( G ) be such that F ^ 2 , 3 , 4 ∈ C c ∞ ⁢ ( ℝ 4 ) . Then there exists φ ∈ C c ∞ ⁢ ( ℝ ) of non-negative values such that, for all t , t ′ , x , x ′ , y , y ′ ∈ ℝ ,

| F ^ 2 , 3 , 4 ⁢ ( t , x , y , 0 ) | ≤ φ ⁢ ( t ) ⁢ φ ⁢ ( x ) ⁢ φ ⁢ ( y ) ,

| F ^ 2 , 3 , 4 ⁢ ( t , x , y , 0 ) - F ^ 2 , 3 , 4 ⁢ ( t ′ , x ′ , y ′ , 0 ) | ≤ | t - t ′ | ⁢ φ ⁢ ( x ) ⁢ φ ⁢ ( y ) + | x - x ′ | ⁢ φ ⁢ ( t ) ⁢ φ ⁢ ( y ) + | y - y ′ | ⁢ φ ⁢ ( t ) ⁢ φ ⁢ ( x ) .

Then it follows that

| ( τ ω k , - ε + ⁢ ( F ) - τ ω k , 0 + ⁢ ( F ) ) ⁢ ( M I k , 1 + ⁢ ( η ) ) ⁢ ( u ) |

= | ∫ I k , 1 + ( F ^ 2 , 3 , 4 ⁢ ( t - ln ⁡ u , ω k ⁢ e - t , - ε ⁢ e t , 0 ) - F ^ 2 , 3 , 4 ⁢ ( t - ln ⁡ u , ω k ⁢ e - t , 0 , 0 ) ) ⁢ M I k , 1 + ⁢ η ⁢ ( e t ) ⁢ 𝑑 t |

≤ ∫ I k , 1 + e t ⁢ φ ⁢ ( t - ln ⁡ ( u ) ) ⁢ φ ⁢ ( ω k ⁢ e - t ) ⁢ | 1 I k , 1 + ⁢ ( e t ) ⁢ η ⁢ ( e t ) | ⁢ 𝑑 t

≤ | ω k | ⁢ Q k ⁢ ∫ I k , 1 + φ ⁢ ( t - ln ⁡ ( u ) ) ⁢ φ ⁢ ( ω k ⁢ e - t ) ⁢ | η ⁢ ( e t ) | ⁢ 𝑑 t .

Hence,

∥ ( τ ω k , - ε + ⁢ ( F ) - τ ω k , 0 + ⁢ ( F ) ) ∘ M I k , 1 + ⁢ ( η ) ∥ 2 ≤ ω k 2 ⁢ Q k 2 ⁢ ∫ ℝ + ( ∫ I k , 1 + φ ⁢ ( t - ln ⁡ ( u ) ) ⁢ φ ⁢ ( ω k ⁢ e - t ) ⁢ | η ⁢ ( e t ) | ⁢ 𝑑 t ) 2 ⁢ d ⁢ u u

≤ ω k 2 ⁢ Q k 2 ⁢ ∫ ℝ + ( ∫ I k , 1 + φ ⁢ ( t - ln ⁡ ( u ) ) ⁢ 𝑑 t ⁢ ∫ I k , 1 + φ ⁢ ( t - ln ⁡ ( u ) ) ⁢ φ ⁢ ( ω k ⁢ e - t ) 2 ⁢ | η ⁢ ( e t ) | 2 ⁢ 𝑑 t ) ⁢ d ⁢ u u

≤ ω k 2 ⁢ Q k 2 ⁢ ∥ φ ∥ 1 ⁢ ∥ φ ∥ ∞ 2 ⁢ ∫ ℝ + ∫ I k , 1 + φ ⁢ ( t - ln ⁡ ( u ) ) ⁢ | η ⁢ ( e t ) | 2 ⁢ 𝑑 t ⁢ d ⁢ u u

≤ ω k 2 ⁢ Q k 2 ⁢ ∥ φ ∥ ∞ 2 ⁢ ∥ φ ∥ 1 2 ⁢ ∥ η ∥ 2 .

Since

0 ≤ ω k ⁢ Q k ≤ ω k ⁢ P k = | λ k | ⁢ | ρ k | ⁢ P k → 0 ,

it follows that

lim k → ∞ ⁡ ∥ τ ω k , - ε + ⁢ ( F ) ∘ M I k , 1 + - τ ω k , 0 + ⁢ ( F ) ∘ M I k , 1 + ∥ op = 0 .

Since for any e t ∈ I k , 2 + we have that

1 P k ≤ | ω k | e t ≤ 1 Q k ,

we get

| ( τ ω k , - ε + ⁢ ( F ) - τ 0 , 0 + ⁢ ( F ) ) ⁢ ( M I k , 2 + ⁢ ( η ) ) ⁢ ( u ) |

= | ∫ I k , 2 + ( F ^ 2 , 3 , 4 ⁢ ( t - ln ⁡ u , ω k ⁢ e - t , - ε ⁢ e t , 0 ) - F ^ 2 , 3 , 4 ⁢ ( t - ln ⁡ u , 0 , 0 , 0 ) ) ⁢ η ⁢ ( e t ) ⁢ 𝑑 t |

≤ ∫ I k , 2 + 1 I k , 2 + ⁢ ( ω k ⁢ e - t ) ⁢ | ω k ⁢ e - t | ⁢ φ ⁢ ( t - ln ⁡ ( u ) ) ⁢ φ ⁢ ( - ε ⁢ e t ) ⁢ | η ⁢ ( e t ) | ⁢ 𝑑 t + ∫ I k , 2 + 1 I k , 2 + ⁢ ( - ε ⁢ e t ) ⁢ e t ⁢ φ ⁢ ( t - ln ⁡ ( u ) ) ⁢ φ ⁢ ( ω k ⁢ e - t ) ⁢ | η ⁢ ( e t ) | ⁢ 𝑑 t

≤ ( 1 Q k + | ω k | ⁢ P k ) ⁢ ∥ φ ∥ ∞ ⁢ ∫ ℝ + φ ⁢ ( t - ln ⁡ ( u ) ) ⁢ | η ⁢ ( e t ) | ⁢ 𝑑 t .

This relation implies that

lim k → ∞ ⁡ ∥ τ ω k , - ε + ⁢ ( F ) ∘ M I k , 2 + - τ 0 , 0 + ⁢ ( F ) ∘ M I k , 2 + ∥ op = 0 .

In the same way, we have that

| ( τ ω k , - ε + ⁢ ( F ) - τ 0 , - ε ⁢ | ω k | + ⁢ ( F ) ) ⁢ ( M I k , 3 + ⁢ ( η ) ) ⁢ ( u ) |

= | ∫ ℝ ( F ^ 2 , 3 , 4 ⁢ ( t - ln ⁡ u , ω k ⁢ e - t , - ε ⁢ e t , 0 ) - F ^ 2 , 3 , 4 ⁢ ( t - ln ⁡ u , 0 , - ε ⁢ | ω k | ⁢ e t , 0 ) ) ⁢ M I k , 3 + ⁢ η ⁢ ( e t ) ⁢ 𝑑 t |

= | ∫ I k , 3 + ( F ^ 2 , 3 , 4 ( t + ln ( | ω k | ) - ln u , 1 sign ⁡ ( ω k ) ⁢ e t , - ε | ω k | e t , 0 )

- F ^ 2 , 3 , 4 ( t + ln ( | ω k | ) - ln u , 0 , - ε | ω k | e t , 0 ) ) η ( | ω k | e t ) d t |

≤ ∫ I k , 3 + e - t ⁢ φ ⁢ ( t + ln ⁡ ( | ω k | ) - ln ⁡ ( u ) ) ⁢ φ ⁢ ( - ε ⁢ | ω k | ⁢ e t ) ⁢ | η ⁢ ( | ω k | ⁢ e t ) | ⁢ 𝑑 t

≤ ∥ φ ∥ ∞ P k ⁢ ∫ I k , 3 + φ ⁢ ( t + ln ⁡ ( | ω k | ) - ln ⁡ ( u ) ) ⁢ | η ⁢ ( | ω k | ⁢ e t ) | ⁢ 𝑑 t .

Hence,

∥ ( τ ω k , - ε + ⁢ ( F ) - τ 0 , - ε ⁢ | ω k | + ⁢ ( F ) ) ∘ M I k , 3 + ⁢ ( η ) ∥ 2

≤ ∥ φ ∥ ∞ 2 P k 2 ⁢ ∫ ℝ + ( ∫ I k , 3 + φ ⁢ ( t + ln ⁡ ( | ω k | ) - ln ⁡ ( u ) ) ⁢ M I k , 3 + ⁢ η ⁢ ( | ω k | ⁢ e t ) ⁢ 𝑑 t ) 2 ⁢ d ⁢ u u

≤ ∥ φ ∥ ∞ 2 P k 2 ⁢ ∫ ℝ + ( ∫ I k , 3 + φ ⁢ ( t + ln ⁡ ( | ω k | ) - ln ⁡ ( u ) ) ⁢ 𝑑 t ⁢ ∫ I k , 3 + φ ⁢ ( t + ln ⁡ ( | ω k | ) - ln ⁡ ( u ) ) ⁢ | η ⁢ ( | ω k | ⁢ e t ) | 2 ⁢ 𝑑 t ) ⁢ d ⁢ u u

≤ ∥ φ ∥ ∞ 2 ⁢ ∥ φ ∥ 1 P k 2 ⁢ ∫ ℝ + ∫ I k , 3 + φ ⁢ ( t + ln ⁡ ( | ω k | ) - ln ⁡ ( u ) ) ⁢ | η ⁢ ( | ω k | ⁢ e t ) | 2 ⁢ 𝑑 t ⁢ d ⁢ u u

≤ ∥ φ ∥ ∞ 2 ⁢ ∥ φ ∥ 1 2 P k 2 ⁢ ∥ η ∥ 2 .

We proceed similarly for the second relation. ∎

Definition 4.14.

Let ( π ρ k , λ k ) k ∈ ℕ be a properly converging sequence with limit set L = Γ 1 ∪ Γ 0 (i.e. lim k → ∞ ⁡ λ k = 0 and lim k → ∞ ⁡ ω k = 0 , where ω k := ρ k ⁢ λ k = ρ k ⁢ ε ⁢ | λ k | ) and ρ k ≠ 0 for k ∈ ℕ . We choose a sequence ( R k ) k ⊂ ℝ + such that

lim k → ∞ ⁡ R k = + ∞ , lim k → ∞ ⁡ R k 2 ⁢ λ k = 0 , lim k → ∞ ⁡ ω k R k 2 ⁢ λ k = 0 ,

and sequences ( Q k ) k , ( P k ) k as in Definition 4.12. For k ∈ ℕ and ϕ ∈ l ∞ ⁢ ( G ^ ) , define the bounded linear operator σ k 0 , ω k , ε ⁢ ( ϕ | L ) (resp. ( σ k 0 , ω k , ε ) ′ ⁢ ( ϕ | L ) ) on L 2 ⁢ ( ℝ ) by

σ k 0 , ω k , ε ⁢ ( ϕ | L ) := V k ∘ s k 0 , ω k , ε , + ⁢ ( ϕ | L ) ∘ V k * ⊕ V k ∘ s k 0 , ω k , ε , - ⁢ ( ϕ | L ) ∘ V k * ,

where

s k 0 , ω k , ε , + ⁢ ( ϕ | L ) := ϕ ⁢ ( τ ω k , 0 + ) ∘ M J k + ⊕ ϕ ⁢ ( τ 0 , 0 + ) ∘ M I k , 2 + ⊕ ϕ ⁢ ( τ 0 , - ε ⁢ | ω k | + ) ∘ M I k , 3 +

and

s k 0 , ω k , ε , - ⁢ ( ϕ | L ) := ϕ ⁢ ( τ - ω k , 0 - ) ∘ M J k - ⊕ ϕ ⁢ ( τ 0 , 0 - ) ∘ M I k , 2 - ⊕ ϕ ⁢ ( τ 0 , ε ⁢ | ω k | - ) ∘ M I k , 3 - .

Theorem 4.15.

Suppose that lim k → ∞ ⁡ λ k = 0 , lim k → ∞ ⁡ ρ k ⁢ λ k = 0 and λ k = ε ⁢ | λ k | for k ∈ N . Then we have

lim k → ∞ ⁡ ∥ π ρ k , λ k ⁢ ( a ) - σ k 0 , ε ⁢ ω k , ε ⁢ ( a ^ | L ) ∥ op = 0

for every a ∈ C * ⁢ ( G ) .

Proof.

Apply Lemma 4.10 and Lemma 4.13. ∎

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 + , d ⁢ x | x | ) . Following [10, Definition 6.2], we define the compact condition on l ∞ ⁢ ( G ^ ) , which will be satisfied by the elements of C * ⁢ ( G ) .

Choose a positive-valued function q ∈ C c ⁢ ( ℝ 2 ) such that ∫ ℝ 2 q ⁢ ( x , y ) ⁢ 𝑑 x ⁢ 𝑑 y = 1 . For f ∈ L 1 ⁢ ( ℝ ) , define the function E ⁢ ( f ) ∈ L 1 ⁢ ( ℝ 3 ) ≃ L 1 ⁢ ( G / 𝒵 ) by

E ⁢ ( f ) ⁢ ( t , x , y ) := f ⁢ ( t ) ⁢ q ⁢ ( x , y ) for all ⁢ ( t , x , y ) ∈ ℝ 3 .

We see that, for any ℓ = ( 0 , μ , ν , 0 ) , μ , ν ∈ ℝ and π μ , ν := π ℓ = π ( 0 , μ , ν , 0 ) ,

π μ , ν ⁢ ( E ⁢ ( f ) ) ⁢ ( η ) ⁢ ( u ) = ∫ ℝ f ⁢ ( u - t ) ⁢ q ^ ⁢ ( μ ⁢ e - t , ν ⁢ e t ) ⁢ η ⁢ ( t ) ⁢ 𝑑 t , η ∈ L 2 ⁢ ( ℝ ) , u ∈ ℝ .

Let l ⁢ ( f ) be the operator

( l ⁢ ( f ) ⁢ ξ ) ⁢ ( u ) := ∫ ℝ f ⁢ ( t ) ⁢ ξ ⁢ ( u - t ) ⁢ 𝑑 t for ⁢ ξ ∈ L 2 ⁢ ( ℝ ) ⁢ and ⁢ u ∈ ℝ ,

given by the regular representation of the group ( ℝ , + ) on the Hilbert space L 2 ⁢ ( ℝ ) , and let q μ , ν be the continuous bounded function defined by

q μ , ν ⁢ ( t ) := q ^ ⁢ ( e - t ⁢ μ , e t ⁢ ν ) for ⁢ t ∈ ℝ .

We thus have

π μ , ν ⁢ ( E ⁢ ( f ) ) ⁢ ( η ) = l ⁢ ( f ) ⁢ ( q μ , ν ⋅ η ) .

Hence,

∥ π μ , ν ⁢ ( E ⁢ ( f ) ) ∥ op ≤ ∥ l ⁢ ( f ) ∥ op ⁢ ∥ q μ , ν ∥ ∞

is bounded for ℓ ∈ Γ 2 ∪ Γ 1 . Clearly, π ℓ ⁢ ( E ⁢ ( f ) ) is bounded for ℓ = ( τ , 0 , 0 , 0 ) ∈ Γ 0 . This shows that the mapping E extends to a bounded linear mapping of C * ⁢ ( ℝ ) into C * ⁢ ( G / 𝒵 ) . We then have the bounded linear mapping

(4.18) σ 0 : C 0 ⁢ ( ℝ ) → C * ⁢ ( G / 𝒵 ) given by σ 0 ⁢ ( ψ ) := E ⁢ ( ℱ - 1 ⁢ ( ψ ) ) ,

where ℱ - 1 : C 0 ⁢ ( ℝ ) → C * ⁢ ( ℝ ) is the inverse Fourier transform.

It follows from Proposition 4.2 that C 0 ⁢ ( Γ 0 ) = C 0 ⁢ ( ℝ ) . Thus, we can apply the bounded linear map σ 0 to ϕ | Γ 0 for any ϕ ∈ l ∞ ⁢ ( G ^ ) .

Definition 4.16.

An operator field ϕ defined on G ^ is said to satisfy the compact condition if, for ε ∈ { + 1 , - 1 } , the operators ϕ ⁢ ( π ε , 0 ) - π ε , 0 ⁢ ( σ 0 ⁢ ( ϕ | Γ 0 ) ) and ϕ ⁢ ( π 0 , ε ) - π 0 , ε ⁢ ( σ 0 ⁢ ( ϕ | Γ 0 ) ) are compact. Here π ε , 0 (resp. π 0 , ε ) is the representation π ℓ , where ℓ = ( 0 , ε , 0 , 0 ) (resp. ℓ = ( 0 , 0 , ε , 0 ) ).

Let ( O ℓ k ) k , ℓ k = ( 0 , ε ⁢ ρ k , σ , 0 ) ∈ Γ 2 for all k, be a properly converging sequence in G ^ , whose limit set contains the orbits O ( 0 , ε , 0 , 0 ) and O ( 0 , 0 , σ , 0 ) . Let ( r k ) k ⊂ ℝ be such that e - r k ⁢ ρ k = 1 for all k ∈ ℕ . Then lim k → ∞ ⁡ r k = - ∞ . Choose a positive sequence ( α k ) k such that α k > - r k for all k ∈ ℕ ,

lim k → ∞ ⁡ α k + r k = ∞ and lim k → ∞ ⁡ α k + r k r k = 0 .

We say that the sequence ( α k ) k is adapted to the sequence ( ℓ k ) k . For r ∈ ℝ , let U ⁢ ( r ) be the unitary operator on L 2 ⁢ ( ℝ ) defined by

(4.19) U ⁢ ( r ) ⁢ ξ ⁢ ( s ) := ξ ⁢ ( s + r ) for all ⁢ ξ ∈ L 2 ⁢ ( ℝ ) ⁢ and ⁢ s ∈ ℝ .

Definition 4.17.

Let ϕ be an operator field defined over G ^ . We say that ϕ satisfies the generic condition (see [10, Definition 5.10]) if, for every properly converging sequence ( π ℓ k ) k ⊂ Γ 2 which admits limit points π ( 0 , ε , 0 , 0 ) , π ( 0 , 0 , σ , 0 ) and for a sequence ( α k ) k adapted to the sequence ( ℓ k ) k , we have that

lim k → ∞ ⁡ ∥ U ⁢ ( r k ) ∘ ϕ ⁢ ( π ℓ k ) ∘ U ⁢ ( - r k ) ∘ M ( - ∞ , α k ) - ϕ ⁢ ( π 0 , ε , 0 , 0 ) ∘ M ( - ∞ , α k ) ∥ op = 0 ,

lim k → ∞ ∥ U ( r k ) ∘ ϕ ( π ℓ k ) ∘ U ( - r k ) ∘ M ( - α k , ∞ ) - ϕ ( π 0 , 0 , σ , 0 ) ∘ M ( - α k , ∞ ) ∥ op = 0 .

Remark 4.18.

According to [10, Proposition 5.12], every ϕ ∈ C * ⁢ ( G / 𝒵 ) ^ satisfies the generic condition as G / 𝒵 is an a ⁢ x + b -like group.

We identify as before the spectrum G ^ of G with the set

Γ = Σ 3 ∪ Σ 2 ∪ Σ 1 ∪ Σ 0 ,

and the C*-algebra l ∞ ⁢ ( G ^ ) is then the algebra of all uniformly bounded operator fields ϕ defined on G ^ with values in B ⁢ ( L 2 ⁢ ( ℝ ) ) on Σ 3 , values in ℬ ⁢ ( L 2 ⁢ ( ℝ + , d ⁢ x | x | ) ) on Σ 2 ∪ Σ 1 , and with values in ℂ on Σ 0 . In the following, we define the subset D * ⁢ ( G ) of l ∞ ⁢ ( G ^ ) , which will be our desired C*-algebra of Boidol’s group.

Definition 4.19.

Let D * ⁢ ( G ) be the subset of l ∞ ⁢ ( G ^ ) consisting of all operator fields ϕ defined over G ^ with the following properties:

The mapping γ ↦ ϕ ⁢ ( γ ) vanishes at infinity.
1. The mapping γ ↦ ϕ ⁢ ( π γ ) is norm continuous on the set Σ 3 .
2. For any γ ∈ Σ 3 , the operator ϕ ⁢ ( γ ) is compact.
3. For every properly converging sequence ( π ρ k , λ k ) k in Σ 3 with limit set L ∈ Σ 2 , i.e. when
  
  lim k → ∞ ⁡ ω k = ω ≠ 0 , where ⁢ ω k = ρ k ⁢ λ k ⁢ and ⁢ λ k = ε ⁢ | λ k | ,
  
  for the sequence ( R k ) k ⊂ ℝ + with the properties given in (4.5) and the operator σ k ω defined in (4.11), we have that
  
  lim k → ∞ ⁡ ∥ ϕ ⁢ ( π ρ k , λ k ) - σ k ω ⁢ ( ϕ | L ) ∥ op = 0 .
4. For every properly converging sequence ( π ρ k , λ k ) k in Σ 3 with limit set L = Σ 1 ∪ Σ 0 , i.e. when
  
  lim k → ∞ ⁡ ω k = 0 ,
  
  admitting a sub-sequence, we have
  
  lim k → ∞ ⁡ ∥ ϕ ⁢ ( π ρ k , λ k ) - σ k 0 , ε ⁢ ω k , ε ⁢ ( ϕ | L ) ∥ op = 0 ,
  
  where the sequences ( R k ) , ( P k ) and ( Q k ) are as in Definition 4.12.
1. The mappings γ ↦ ϕ ⁢ ( π γ ) are norm continuous on the sets Σ j for j = 0 , 1 , 2 (in particular the function ϕ | Σ 0 is in C 0 ⁢ ( Σ 0 ) = C 0 ⁢ ( ℝ ) ≃ C * ⁢ ( ℝ ) ).
2. For any γ ∈ Σ 2 , the operator ϕ ⁢ ( γ ) is compact.
3. For every properly converging sequence ( τ ω k , - ε ) k in Σ 2 , ε ∈ { + 1 , - 1 } , with limit set L = Σ 1 ∪ Σ 0 (i.e. lim k → ∞ ⁡ ω k = 0 ), for the sequence ( R k ) ⊂ ℝ + with the properties in Definition 4.14, we have that
  
  lim k → ∞ ⁡ ∥ ϕ ⁢ ( τ ω k , - ε + ) - s k 0 , ω k , ε , + ⁢ ( ϕ | L ) ∥ op = 0 ,
  
  lim k → ∞ ⁡ ∥ ϕ ⁢ ( τ - ω k , ε - ) - s k 0 , ω k , ε , - ⁢ ( ϕ | L ) ∥ op = 0 .
4. The operator ϕ ⁢ ( γ ) , γ ∈ Σ 2 , satisfies the generic condition given in Definition 4.17.
5. The operator ϕ ⁢ ( γ ) , γ ∈ Σ 1 , satisfies the compact condition given in Definition 4.16.
The adjoint operator field ϕ * satisfies the same conditions.

Remark 4.20.

The operators σ k 0 , ε ⁢ ω k , ε , s k 0 , ω k , ε , ± and σ k ω are defined in Definition 4.14 and (4.11), respectively. Note that, due to the fact that 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 to the internals I k , 2 ± , J k ± used in Definition 4.14.

We will show that the set D * ⁢ ( G ) defined in Definition 4.19 is a C*-subalgebra of l ∞ ⁢ ( G ^ ) .

Remark 4.21.

Since 𝒵 is the center 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 a ⁢ x + 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 conditions (i), (iii) and (iv) 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 (ii)). 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 ^ | S 2 . 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 S 2 (given in Remark 2.1 (ii)) and the fact that the spectrum C * ⁢ ( G ) ^ can be identified with ⋃ i = 0 3 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 ) = a ^ , a ∈ C * ⁢ ( G ) , satisfies all conditions in D * ⁢ ( G ) . Thus, C * ⁢ ( G ) ^ ⊂ D * ⁢ ( G ) .

Lemma 4.22.

The set D * ⁢ ( G ) is a complete subspace of l ∞ ⁢ ( G ^ ) .

Proof.

Let ( ϕ n ) n ⊂ D * ⁢ ( G ) be a Cauchy sequence. Then, for every γ ∈ G ^ , the sequence ( ϕ n ⁢ ( γ ) ) n is a Cauchy sequence in the C*-algebra of bounded linear operators on the corresponding Hilbert spaces ℋ γ , and so ϕ ⁢ ( γ ) = lim k → ∞ ⁡ ϕ k ⁢ ( γ ) exists in ℬ ⁢ ( ℋ γ ) . By Remark 4.21, there is an operator field ϕ such that ϕ | S 2 is contained in C * ⁢ ( G / 𝒵 ) , and

∥ ϕ n ⁢ ( γ ) - ϕ ⁢ ( γ ) ∥ op → 0 as ⁢ n → ∞

for every γ ∈ S 2 . It then suffices to show that ϕ satisfies conditions (ii) in Definition 4.19. It is clear that ϕ satisfies (ii) (a) and (ii) (b). For any properly converging sequence ( π ρ k , λ k ) k in Σ 3 with limit set in Σ 2 , we have that

lim k → ∞ ⁡ ∥ ϕ n ⁢ ( π ρ k , λ k ) - σ k ω ⁢ ( ϕ n ) ∥ op = 0 for all ⁢ n ,

since ϕ n is in D * ⁢ ( G ) . Hence,

∥ ϕ ⁢ ( π ρ k , λ k ) - σ k ω ⁢ ( ϕ | L ) ∥ op ≤ ∥ ϕ ⁢ ( π ρ k , λ k ) - ϕ n ⁢ ( π ρ k , λ k ) ∥ op + ∥ ϕ n ⁢ ( π ρ k , λ k ) - σ k ω ⁢ ( ϕ n | L ) ∥ op + ∥ σ k ω ⁢ ( ϕ n | L ) - σ k ω ⁢ ( ϕ | L ) ∥ op ,

which converges to zero. Thus, (ii) (c) is satisfied by the operator field ϕ. Similar arguments hold for condition (ii) (d). Therefore, we have that ϕ ∈ D * ⁢ ( G ) . ∎

Lemma 4.23.

The subspace D * ⁢ ( G ) of l ∞ ⁢ ( G ^ ) is a postliminal C*-algebra with spectrum equal to G ^ .

Proof.

It is shown in Lemma 4.22 that D * ⁢ ( G ) is a closed subspace of l ∞ ⁢ ( G ^ ) and it is clear that D * ⁢ ( G ) is involutive. Let us show that it is also a subalgebra. Let ϕ , ϕ ′ be two elements of D * ⁢ ( G ) . By Remark 4.21, there exist a , a ′ ∈ C * ⁢ ( G / 𝒵 ) such that ϕ | S 2 = a ^ and ϕ | S 2 ′ = a ^ ′ . Therefore, we only need to check whether conditions (ii) (b), (ii) (c) and (ii) (d) work for the product ϕ ∘ ϕ ′ . Condition (ii) (b) is evident. Now for any ω ∈ ℝ * , by Theorem 4.7, we have that

lim k → ∞ ⁡ ∥ σ k ω ⁢ ( ( a ⋅ a ′ ^ ) | L ) - σ k ω ⁢ ( a ^ | L ) ∘ σ k ω ⁢ ( a ^ | L ′ ) ∥ op = 0 ,

and so

lim k → ∞ ⁡ ∥ ϕ ∘ ϕ ′ ⁢ ( π ρ k , λ k ) - σ k ω ⁢ ( ϕ ∘ ϕ | L ′ ) ∥ op = lim k → ∞ ⁡ ∥ ϕ ∘ ϕ ′ ⁢ ( π ρ k , λ k ) - σ k ω ⁢ ( ( a ⋅ a ′ ^ ) | L ) ∥ op

= lim k → ∞ ⁡ ∥ ϕ ⁢ ( π ρ k , λ k ) ∘ ϕ ′ ⁢ ( π ρ k , λ k ) - σ k ω ⁢ ( a ^ | L ) ∘ σ k ω ⁢ ( a ^ | L ′ ) ∥ op

= 0 .

Condition (ii) (d) follows in a similar way thanks to Theorem 4.15.

Hence, D * ⁢ ( G ) , being a closed * -subalgebra of l ∞ ⁢ ( G ^ ) , 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 ) ), 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 S 2 .

It follows immediately from conditions (i) and (ii) that the ideal K 0 of D * ⁢ ( G ) is just the algebra C 0 ⁢ ( Γ 3 , 𝒦 ) of continuous mappings defined on the locally compact space Γ 3 with values in the algebra 𝒦 of compact operators on L 2 ⁢ ( ℝ ) . If π ⁢ ( K 0 ) = ( 0 ) , then π can be identified with an irreducible representation of C * ⁢ ( G / 𝒵 ) , so there exists π ′ ∈ S 2 such that π ⁢ ( ϕ ) = ϕ ⁢ ( π ′ ) for ϕ ∈ D * ⁢ ( G ) . If π ⁢ ( K 0 ) ≠ ( 0 ) , then π defines an irreducible representation of the algebra C 0 ⁢ ( Γ 3 , 𝒦 ) , and hence there exists π ρ , λ ∈ Σ 3 such that π ⁢ ( ϕ ) = ϕ ⁢ ( π ρ , λ ) for ϕ ∈ K 0 , and finally π is the evaluation at π ρ , λ for every ϕ ∈ D * ⁢ ( G ) .

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 (ii) (b), (iii) (b), (iii) (e)). ∎

Now we have our main theorem, which characterizes 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 (3.15) 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 [4, 11.1.8]) tells us that the subalgebra C * ⁢ ( G ) ^ of D * ⁢ ( G ) is equal to D * ⁢ ( G ) . ∎

Communicated by Siegfried Echterhoff

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Received: 2021-08-13

Revised: 2023-12-19

Published Online: 2024-02-01

Published in Print: 2024-07-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/forum-2021-0209

Keywords for this article

C*-algebra; algebra of operator fields; spectrum

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