The $C^*$-algebra of the Cartan motion groups

TL;DR

This paper characterizes the $C^*$-algebra of Cartan motion groups by describing it through operator fields, providing a detailed structural understanding under certain assumptions.

Contribution

It offers a novel description of the $C^*$-algebra of Cartan motion groups in terms of operator fields, expanding the theoretical framework for these groups.

Findings

Explicit description of $C^*(G_0)$ in terms of operator fields

Structural insights into the $C^*$-algebra of Cartan motion groups

Conditions under which the description holds

Abstract

Let $G_0=K\ltimes\p$ be the Cartan motion groups. Under some assumption on $G_0,$ we describe the $C^*$-algebra $C^*(G_0)$ of $G_0$ in terms of operator fields.

Equations

$$\Vert f{\Vert }_{C^{\ast }(G)}:=\underset{\pi \in \hat{G}}{\sup }\Vert \pi (f){\Vert }_{op}.$$

$$\widehat{C^{\ast }(G)}\simeq \hat{G}.$$

$$\ell^{\infty}(\widehat{G}):=\Big{\{}F:\widehat{G}\longrightarrow\bigcup_{\pi\in\widehat{G}}\mathcal{B}(\mathcal{H}_{\pi}),\,\pi\longmapsto F(\pi)\in\mathcal{B}(\mathcal{H}_{\pi});\,\|F\|_{\infty}:=\sup_{\pi\in\widehat{G}}\|F(\pi)\|_{op}<\infty\Big{\}}$$

$$\mathcal{F}(f)(\pi )=\pi (f),\pi \in \hat{G},f\in C^{\ast }(G).$$

$$(k_1,X_1)\cdot (k_2,X_2)=(k_1{k}_2,X_1+Ad(k_1)X_2),(k_1,k_2)\in K,(X_1,X_2)\in \mathfrak{p}.$$

$$W:=N_K(\mathfrak{a})/Z_K(\mathfrak{a})$$

$${\tilde{C}}^{+}(\tilde{\mathfrak{a}})=\{\tilde{H}\in \tilde{\mathfrak{a}};\alpha (\tilde{H})>0,\forall \alpha \in {\Sigma }^{+}\}.$$

$$\rho \otimes {\chi }_{\phi }:(k,X)⟼e^{i\phi (X)}\rho (k)$$

$$f(k{k}_0)=\rho (k_0^{-1})f(k)$$

$${\pi }_{(\rho ,\phi )}:=In{d}_{K_{\phi }⋉\mathfrak{p}}^{G_0}(\rho \otimes {\chi }_{\phi })$$

$${\pi }_{(\rho ,\phi )}(k_0,X)f(k)=e^{i\phi (Ad(k^{-1})X)}f(k_0^{-1}k),$$

$${\Gamma }_0\simeq \hat{M}\times C^{+}(\mathfrak{a}).$$

$${\Gamma }_0\simeq \hat{M}\times {\mathbb{R}}_{+}^{\ast }\text{and}\widehat{G_0}\simeq {\Gamma }_0\bigsqcup {\Gamma }_2$$

$$\Gamma_{1}:=\Big{\{}\pi_{(\mu,H)}=Ind_{K_{\varphi_{H}}\ltimes\mathfrak{p}}^{K\ltimes\mathfrak{p}}(\rho\otimes\chi_{\varphi_{H}}),\,\,\rho\in\widehat{K_{\varphi_{H}}},\,H\in\partial(C^{+}(\mathfrak{a}))\backslash\{0\}\Big{\}}$$

$$\widehat{G_0}\simeq {\Gamma }_0\bigsqcup {\Gamma }_1\bigsqcup {\Gamma }_2.$$

$$K_{\Lambda }⋉\mathfrak{p}$$

$$(k,X)$$

$$|{\beta }_1+\sqrt{-1}{\beta }_2{|}^2=B({\beta }_1,{\beta }_1)+B({\beta }_2,{\beta }_2)$$

$$\mathcal{U}:=\Big{\{}(\rho^{{}^{\prime}},\Lambda^{{}^{\prime}})\in\mathcal{F}_{c}:\,|\Lambda-\Lambda^{{}^{\prime}}|<\varepsilon\,\,and\,\,[\rho\big{|}_{K_{\Lambda^{{}^{\prime}}}}:\rho^{{}^{\prime}}]>0\Big{\}}$$

$$w.(\rho ,\Lambda )=(w.\rho ,w.\Lambda ).$$

$$\mathcal{F}:=\Big{\{}(\rho,\Lambda)\in\mathcal{F}_{c}:\,\Lambda=\sqrt{-1}\beta\,\,where\,\,\beta\,\,is\,\,real\,\,valued\Big{\}}.$$

$$\mathcal{F}/W$$

$$(\rho ,\Lambda )$$

$${\Lambda }_H(X)=\sqrt{-1}⟨H,X⟩\forall X\in \mathfrak{a}.$$

$$\mathcal{V}:=\Big{\{}(\rho^{{}^{\prime}},\Lambda^{{}^{\prime}})\in\mathcal{F}_{c}:\,|\Lambda_{H}-\Lambda^{{}^{\prime}}|<\varepsilon\,\,and\,\,[\rho_{\mu}\big{|}_{K_{\Lambda^{{}^{\prime}}}}:\rho^{{}^{\prime}}]>0\Big{\}}$$

$$\forall n\ge n_0,({\rho }_{{\mu }^n},{\Lambda }_{H_n})\in \mathcal{V}.$$

$$|{\Lambda }_{H_n}-{\Lambda }_H|$$

$$\displaystyle[\rho_{\mu}\big{|}_{K_{H_{n}}}:\rho_{\mu^{n}}]$$

$$H_n$$

$$\displaystyle[\rho_{\mu}\big{|}_{K_{H_{n}}}:\rho_{\mu^{n}}]$$

$$[\rho_{\mu}\big{|}_{K_{H_{n}}}:\rho_{\mu^{n}}]>0\Longleftrightarrow[\rho_{\mu}\big{|}_{M}:\rho_{\mu^{n}}]>0$$

Full text

1. Introduction

Let $G$ be a locally compact group. We denote by $\widehat{G}$ the unitary dual of $G.$ It well-known that $\widehat{G}$ equipped with the Fell topology (see [18, 19]). The first representation-theoretic question concerning the group $G$ is the full parametrezation and topological identification of the dual $\widehat{G}.$ The $C^{*}$-algebra $C^{*}(G)$ is the completion of the convolution algebra $L^{1}(G)$ equipped with the $C^{*}$-norm $\|.\|_{C^{*}(G)}$, given by

[TABLE]

We denote by $\widehat{C^{*}(G)}$ the unitary dual of the $C^{*}$-algebra of $G.$ Then we have the following bijection

[TABLE]

Furthermore, the $C^{*}$-algebra $C^{*}(G)$ of $G$ can be identified with a subalgebra of the large $C^{*}$-algebra $\ell^{\infty}(\widehat{G})$ of bounded operator fields given by

[TABLE]

under the Fourier transform $\mathcal{F}$ defined on $C^{*}(G)$ as follows:

[TABLE]

Using the fact that $\mathcal{F}$ is an injective homomorphism of $C^{*}(G)$ into $\ell^{\infty}(\widehat{G}),$ then the $C^{*}$-algebra $C^{*}(G)$ is isomorphic to a subalgebra $\mathcal{D}:=\mathcal{F}(C^{*}(G))$ of elements in $\ell^{\infty}(\widehat{G})$ verifying some conditions. The elements of $\mathcal{D}$ must naturally fulfil is that of continuity. Then the parametrization and the descreption of the topology of $\widehat{G}$ are required to describe the $C^{*}$-algebra $C^{*}(G)$ of $G.$

In this context, we have some works in the literature , for example, J. Ludwig and L. Turowska have described in [16] the $C^{*}$-algebra of the Heisenberg group and of the thread-like Lie groups in terms of an algebra of operator fields defined over their dual spaces. The descreption of the $C^{*}$-algebra of the Euclidean motion group $M_{n}:=SO(n)\ltimes\mathbb{R}^{n},\,n\in\mathbb{N}^{*}$ was established in [1]. In the present work, we give a similar precise descreption of the $C^{*}$-algebra of the Cartan motion groups. Our result is a generalization of analogous results in the case of the Euclidean motion group (see [1]).

The paper is organized as follows. In section 2, we introduce the groups $G_{0}=K\ltimes\mathfrak{p}$, the semi-direct product of the maximal compact connected subgroup $K$ of same connected semisimple Lie group with finite center $G$, acting by adjoint action on $\mathfrak{p}$ (where $\mathfrak{p}$ determined by the Cartan decomposition of Lie algebra of the group $G$). We recall the topology of the spectrum of the groups $G_{0}$ and we determine the convergence in $\widehat{G_{0}}$. In the last section, we determine the Fourier transform of their group $C^{*}$-algebras and we describe the elements of the image of the Fourier transform of $C^{*}(G_{0})$ inside the big algebra $\ell^{\infty}(\widehat{G_{0}}).$ This is the main result of the paper

2. The Cartan motion groups $G_{0}$.

Let $G$ be a connected semisimple Lie group with finite center and $K$ a maximal compact connected subgroup of $G.$ Let $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be the corresponding Cartan decomposition of the Lie algebra $\mathfrak{g}$ of $G$ with $\mathfrak{k}:=Lie(K).$ Then one can form the semidirect product $G_{0}:=K\ltimes\mathfrak{p}$ with respect to the adjoint action of $K$ on $\mathfrak{p}.$ The group $G_{0}$ is called the Cartan motion group associated to the Riemannian symmetric pair $(G,K).$ The multiplication in this group is given by

[TABLE]

The group $M_{n}=SO(n)\ltimes\mathbb{R}^{n}$ is an example of Cartan motion groups. More precisely, $M_{n}$ is the Cartan motion group associated to the compact Riemannian symmetric pair $(SO(n+1),SO(n))$.

Let now $\mathfrak{a}$ be a maximal abelian subspace of $\mathfrak{p}$. The dimension of the real vector space $\mathfrak{a}$ is called the rank of the Riemannian symmetric pair $(G,K)$. An important fact worth mentioning here is that every adjoint orbit of $K$ in $\mathfrak{p}$ intersects $\mathfrak{a}$ (see [9] p. 247 ). Let $N_{K}(\mathfrak{a})$ and $Z_{K}(\mathfrak{a})$ denote respectively the normalizer and centralizer of $\mathfrak{a}$ in $K.$ The quotient group

[TABLE]

is called the Weyl group of the pair $(G,K)$. We shall denote the action of $W$ on $\mathfrak{a}$ by $H\longmapsto s.H$ for $H\in\mathfrak{a}$ and $s\in W$. Let us take the subspaces $\tilde{\mathfrak{a}}:=i\mathfrak{a}$, $\tilde{\mathfrak{p}}:=i\mathfrak{p}$ and $\tilde{\mathfrak{g}}:=\mathfrak{k}\oplus\tilde{\mathfrak{p}}$ of the complexification $\mathfrak{g}^{\mathbb{C}}$ of $\mathfrak{g}$. An element $H\in\mathfrak{a}$ is called regular if $\alpha(H)\neq 0$ for all $\alpha\in\Sigma$ where $\Sigma$ is the set of all restricted roots associated to the pair $(\mathfrak{g},\mathfrak{a}).$ A connected component of the set of regular elements in $\mathfrak{a}$ is called a Weyl chamber of the pair $(G,K)$. Endow the dual space $\tilde{\mathfrak{a}}^{*}$ with a lexicographic ordering and denote by $\Sigma^{+}$ the set of positive restricted roots. As an example of Weyl chambers, let us set $C^{+}(\mathfrak{a}):=i\,\widetilde{C}^{+}(\tilde{\mathfrak{a}})$ with

[TABLE]

It is well-known that every $s\in W$ permutes the Weyl chambers and that $W$ acts simply transitively on the set of Weyl chambers. Furthermore, we have the following important result (see [9], p. 322):

Proposition 2.1**.**

Let $C\subset\mathfrak{a}$ be a Weyl chamber. Each orbit of $W$ in $\mathfrak{a}$ intersects the closure $\overline{C}$ in exactly one point.

We shall briefly review the description of the unitary dual of $G_{0}$ via Mackey’s little group theory. Let $\varphi$ be a non-zero linear form on $\mathfrak{p}$. We denote by $\chi_{\varphi}$ the unitary character of the vector Lie group $\mathfrak{p}$ given by $\chi_{\varphi}=e^{i\varphi}$. Let $K_{\varphi}$ be the stabilizer of $\varphi$ under the coadjoint action of $K$ on $\mathfrak{p}^{*}$, and let $\rho$ be an irreducible unitary representation of $K_{\varphi}$ on some Hilbert space $\mathcal{H}_{\rho}$. The map

[TABLE]

is a representation of the semidirect product $K_{\varphi}\ltimes\mathfrak{p}$, which we may induce up so as to obtain a unitary representation of $G_{0}$. Let $L_{\rho}^{2}(K,\mathcal{H}_{\rho})$ be the subspace of $L^{2}(K,\mathcal{H}_{\rho})$ consisting of the maps $f$ which satisfy the covariance condition

[TABLE]

for $k_{0}\in K_{\varphi}$ and $k\in K$. The induced representation

[TABLE]

is realized on $\mathcal{H}_{\rho,\varphi}:=L_{\rho}^{2}(K,\mathcal{H}_{\rho})$ by

[TABLE]

where $(k_{0},X)\in G_{0}$, $f\in L_{\rho}^{2}(K,\mathcal{H}_{\rho})$ and $k\in K$. Mackey’s theory tells us that the representation $\pi_{(\rho,\varphi)}$ is irreducible and that every infinite dimensional irreducible unitary representation of $G_{0}$ is equivalent to some $\pi_{(\rho,\varphi)}$. Furthermore, two representations $\pi_{(\rho,\varphi)}$ and $\pi_{(\rho^{\,{}^{\prime}},\varphi^{\,{}^{\prime}})}$ are equivalent if and only if $\varphi$ and $\varphi^{\,{}^{\prime}}$ lie in the same coadjoint orbit of $K$ and the representations $\rho$ and $\rho^{\,{}^{\prime}}$ are equivalent under the identification of the conjugate subgroups $K_{\varphi}$ and $K_{\varphi^{\,{}^{\prime}}}$. In this way, we obtain all irreducible representations of $G_{0}$ which are not trivial on the normal subgroup $\mathfrak{p}$. On the other hand, every irreducible unitary representation $\tau$ of $K$ extends trivially to an irreducible representation, also denoted by $\tau$, of $G_{0}$ by $\tau(k,X):=\tau(k)$ for $k\in K$ and $X\in\mathfrak{p}$.

Next, we shall provide a more precise description of the so-called “generic irreducible unitary representations” of $G_{0}$. Denote again by $\langle\>,\>\rangle$ the restriction to $\mathfrak{p}\times\mathfrak{p}$ of the $Ad(K)$-invariant scalar product $\langle\>,\>\rangle$ on $\mathfrak{g}_{0}$. Let $\mathfrak{a}$ be a maximal abelian subspace of $\mathfrak{p}$, and let $M$ be the centralizer of $A=exp_{{}_{G}}(\mathfrak{a})$ in $K$. In general, the compact Lie group $M$ is not connected, and one can prove that $M=M_{e}\cdot(M\cap A)$ with $M_{e}$ being the identity component of $M$. A proof of the following well-known result can be found in [10].

Lemma 2.2**.**

Let $C\subset\mathfrak{a}$ be a Weyl chamber. Every adjoint orbit of $K$ in $\mathfrak{p}$ intersects the closure $\overline{C}$ in exactly one point.

We conclude that every infinite dimensional unitary representation of $G_{0}$ has the form $\pi_{(\rho,\varphi_{{}_{H}})}$, where $H$ is a non-zero vector in $\overline{C^{+}(\mathfrak{a})}$ and $\varphi_{{}_{H}}$ is the linear form on $\mathfrak{p}$ given by $\varphi_{{}_{H}}(X)=\langle H,X\rangle$. Observe that the isotropy group $K_{\varphi_{{}_{H}}}$ coincides with the centralizer $Z_{K}(H)$. Let us fix a regular element $H$ in $\mathfrak{a}$. The subgroups $K_{\varphi_{{}_{H}}}$ and $M$ of $K$ are identical. If $\rho$ is an irreducible representation of $M$, then the representation $\pi_{(\rho,\varphi_{{}_{H}})}$ corresponding to the pair $(\rho,\varphi_{{}_{H}})$ is said to be generic. We denote by $\Gamma_{0}$ the set of all equivalence classes of generic irreducible unitary representations of $G_{0}$. Notice that $\Gamma_{0}$ has full Plancherel measure in the unitary dual $\widehat{G_{0}}$. Applying Mackey’s analysis and the result of Lemma 2.3, we obtain the bijection

[TABLE]

Note that when the Riemannian symmetric pair $(G,K)$ has rank one, we can find a unit vector $H_{0}\in\mathfrak{a}$ such that $C^{+}(\mathfrak{a})=\mathbb{R}_{+}^{*}H_{0}$. In this case we have the bijections

[TABLE]

where $\Gamma_{2}:=\widehat{K}.$ Now, we define the subset $\Gamma_{1}\subset\widehat{G_{0}}$ as follows

[TABLE]

where $\partial(C^{+}(\mathfrak{a}))$ is the boundary of $C^{+}(\mathfrak{a}).$ According to Mackey’s theory, we obtain the following parametrization as sets

[TABLE]

We denote by $\mathfrak{z}$ the orthogonal complement of $\mathfrak{a}$ in $\mathfrak{p}$ ($\mathfrak{p}=\mathfrak{a}\oplus\mathfrak{z}$). Let $\Lambda:\mathfrak{a}\longrightarrow\mathbb{C}$ be a real linear function. Also we denote by $\Lambda$ the extension of $\Lambda$ to $\mathfrak{p}$ so that $\mathfrak{z}\subseteq Ker(\Lambda),$ and let $\rho\in\widehat{K_{\Lambda}}.$ We denote by $\pi:=\pi_{(\rho,\Lambda)}$ the representation of $G_{0}$ induced from

[TABLE]

Now, we describe the Fell topology on $\widehat{G_{0}}.$ For $\beta_{1},\beta_{2}\in\mathfrak{a}^{*}$ (the dual vector space of $\mathfrak{a}$), define

[TABLE]

where $B$ is the Cartan-Killing form on $\mathfrak{g}_{0}.$ Let $\mathcal{F}_{c}$ be the set of all pairs $(\rho,\Lambda)$ where $\rho\in\widehat{K}_{\Lambda}.$ We take $(\rho,\Lambda)\in\mathcal{F}_{c},$ if $\varepsilon>0$ is sufficiently small then $|\Lambda-\Lambda^{{}^{\prime}}|<\varepsilon$ implies $K_{\Lambda^{{}^{\prime}}}\subseteq K_{\Lambda}.$ So the subset

[TABLE]

([\rho\big{|}_{K_{\Lambda^{{}^{\prime}}}}:\rho^{{}^{\prime}}] is the multiplicity of $\rho^{{}^{\prime}}$ in \rho\big{|}_{K_{\Lambda}}) defines a basis for the neighborhoods of $(\rho,\Lambda)$ in the topology we give $\mathcal{F}_{c}$ (see [13]). Note that $W$ acts on $\mathcal{F}_{c}$ by

[TABLE]

Let $\mathcal{F}_{c}/W$ be the quotient space by this action of $W,$ equipped with the quotient topology. Now, let

[TABLE]

According to [13], then we have the useful Lemma.

Lemma 2.3**.**

The unitary dual $\widehat{G_{0}}$ of $G_{0}$ is homeomorphic to $\mathcal{F}/W.$

In the remainder of this paper, we shall assume that the stabilizer $K_{\varphi}$ is connected for each $\varphi\in\mathfrak{p}^{*}.$ Let $\rho_{\mu}$ be an irreducible representation of $K_{H}$ with highest weight $\mu$. For simplicity, we shall write $\pi_{(\mu,H)}$ and $\mathcal{H}_{(\mu,H)}$ instead of $\pi_{(\rho_{\mu},\varphi_{{}_{H}})}$ and $\mathcal{H}_{\rho_{\mu},\varphi_{H}}$ respectively. We have:

Proposition 2.4**.**

Let $(\pi_{(\mu^{n},H_{n})})_{n}$ be a sequence in $\Gamma_{0}.$ Then we have:

1. (1)

The sequence $(\pi_{(\mu^{n},H_{n})})_{n}$ converges to $\pi_{(\mu,H)}$ in $\Gamma_{0}$ if and only if $(H_{n})_{n}$ converges to $H$ and $\mu^{n}=\mu$ for $n$ large enough. 2. (2)

The sequence $(\pi_{(\mu^{n},H_{n})})_{n}$ converges to $\pi_{(\mu,H)}$ in $\Gamma_{1}$ if and only if $(H_{n})_{n}$ converges to $H$ and [\rho_{\mu}\big{|}_{M}:\rho_{\mu^{n}}]>0 for $n$ large enough. 3. (3)

The sequence $(\pi_{(\mu^{n},H_{n})})_{n}$ converges to $\tau_{\lambda}$ in $\Gamma_{2}$ if and only if $(H_{n})_{n}$ converges to [math] and [\tau_{\lambda}\big{|}_{M}:\rho_{\mu^{n}}]>0 for $n$ large enough.

Proof.

By Lemma 2.3, we show that the map

[TABLE]

is a homeomorphism (see [13]). Let $H$ be a non-zero vector in $\overline{C^{+}(\mathfrak{a})}$ and $\Lambda_{H}$ is the linear form on $\mathfrak{a}$ given by

[TABLE]

For simplicity, we shall write $K_{H}$ instead of $K_{\Lambda_{H}}.$ Now, we assume that $(H_{n})_{{}_{n}\in I}$ converges to $H$ and [\rho_{\mu}\big{|}_{K_{H_{n}}}:\rho_{\mu^{n}}]>0 for $n$ large enough. This is equivalent that the net $(\rho_{\mu^{n}},\Lambda_{H_{n}})_{n\in I}$ converges to $(\rho_{\mu},\Lambda_{H})$ in $\mathcal{F}/W.$ In view of the continuity of the map $\mathcal{F}/W\ni(\rho,\Lambda)\longmapsto\pi_{(\rho,\Lambda)}\in\widehat{G_{0}},$ we easily see that $(\pi_{(\mu^{n},H_{n})})_{n}$ converges to $\pi_{(\mu,H)}$ in $\widehat{G_{0}}.$

Conversely, assume that the net $(\pi_{(\mu^{n},H_{n})})_{n\in I}$ converges to $\pi_{(\mu,H)}$ in $\widehat{G_{0}}.$ Then the net $((\rho_{\mu^{n}},\Lambda_{H_{n}}))_{n\in I}$ converges to $(\rho_{\mu},\Lambda_{H})$ in $\mathcal{F}/W.$ Recall that the set

[TABLE]

defines a basis for the neighborhoods of $(\rho_{\mu},\Lambda_{H}).$ Hence for each $\varepsilon>0,$ there exists $n_{0}\in I$ such that

[TABLE]

i.e.; $\forall n\geq n_{0},$ we have

[TABLE]

and

[TABLE]

Then we obtain the following

[TABLE]

and

[TABLE]

for $n$ large enough. Notice that for $H,H_{n}\in C^{+}(\mathfrak{a}),$ $K_{H}=K_{H_{n}}=M$ and we get

[TABLE]

for $n$ large enough, which is equivalent to $\mu^{n}=\mu$ for $n$ large enough. This completes the proof of the Proposition. ∎

Proposition 2.5**.**

Let $(\pi_{(\mu^{n},H_{n})})_{n}$ be a sequence in $\Gamma_{1}.$ Then we have:

1. (1)

The sequence $(\pi_{(\mu^{n},H_{n})})_{n}$ converges to $\pi_{(\mu,H)}$ in $\Gamma_{1}$ if and only if $(H_{n})_{n}$ converges to $H$ and [\rho_{\mu}\big{|}_{K_{H_{n}}}:\rho_{\mu^{n}}]>0 for $n$ large enough. 2. (2)

The sequence $(\pi_{(\mu^{n},H_{n})})_{n}$ converges to $\tau_{\lambda}$ in $\Gamma_{2}$ if and only if $(H_{n})_{n}$ converges to [math] and [\tau_{\lambda}\big{|}_{K_{H_{n}}}:\rho_{\mu^{n}}]>0 for $n$ large enough.

Proof.

Applying the same arguments as in the proof of Proposition 2.4.

∎

Of course $\widehat{K}$ has the discrete topology.

Let $C^{*}(G_{0})$ denote the full $C^{*}$-algebra of $G_{0}.$ We denote by $C_{0}(\widehat{K})$ the Banach algebra of all operator fields

[TABLE]

such that $F(\pi)\in\mathcal{B}(\mathcal{H}_{\pi})$ for each irreducible unitary representation $\pi$ of $K$ and such that $\lim_{\pi\longrightarrow\infty}\|F(\pi)\|_{op}=0.$ This algebra is equipped with the norm $\|F\|_{\infty}=\sup_{\pi\in\widehat{K}}\|F(\pi)\|_{op},\,F\in C_{0}(\widehat{K}).$ Its well-known that the $C^{*}$-algebra $C^{*}(K)$ of $K$ is isomorphic to $C_{0}(\widehat{K})$ (see, [1]).

In the sequel, we describe the elements of the image of the Fourier transform of $C^{*}(G_{0})$ inside the big algebra $\ell^{\infty}(\widehat{G_{0}}).$

Definition 2.6**.**

Let $(\mu,H)\in\Gamma_{0}\sqcup\Gamma_{1}$ and let $\rho_{\mu}\in\widehat{K_{H}}.$ We define the representation $\pi_{\mu,0}$ of $G_{0}$ by

[TABLE]

and let $\mathcal{H}_{\mu,0}$ be its Hilbert space. By the Frobenius reciprocity, we obtain

[TABLE]

where $\lambda\geq\mu$ means that $\tau_{\lambda}\in Ind_{K_{H}}^{K}(\rho_{\mu}).$

3. The $C^{*}$-algebra of the group $G_{0}$.

3.1. The Fourier transform.

Now, let $f\in L^{1}(G_{0}),h\in K$ and $\Psi\in\mathcal{H}_{\mu,H},$ then we can give the expression of the operator $\pi_{(\mu,H)}(f)$ by the following equality

[TABLE]

Then

[TABLE]

here $\widehat{f}^{2}$ denotes the partial Fourier transform of $f$ in the second variable and where

[TABLE]

Definition 3.1**.**

Let

[TABLE]

This space $C_{0,2}(G_{0})$ is dense in $L^{1}(G_{0})$ and hence also in $C^{*}(G_{0}).$

Definition 3.2**.**

For each $f\in C^{*}(G_{0}),$ the Fourier transform $\mathcal{F}(f)$ of $f$ is an isometric homomorphism on $C^{*}(G_{0})$ into $\ell^{\infty}(\widehat{G_{0}})$ which is given by

[TABLE]

In the following Proposition, we give a description of elements $\mathcal{F}(f)$ for each $f\in C^{*}(G_{0}).$

Proposition 3.3**.**

Let $f\in C^{*}(G_{0}).$ Then we have:

1. (1)

For every $(\mu,H)\in\Gamma_{0}\sqcup\Gamma_{1},$ ${\mathcal{F}}(\mu,H)$ is a compact operator on $\mathcal{H}_{\mu,H}.$ 2. (2)

The mapping

[TABLE]

are norm continuous on the difference sets $\Gamma_{i},\ i=0,\ 1,\ 2$.. 3. (3)

$\underset{|\mu|\longrightarrow+\infty}{\lim}\|\mathcal{F}(f)(\mu,H)\|_{op}=0$* .* 4. (4)

$\underset{H\longrightarrow 0}{\lim}\|\pi_{(\mu,H)}(f)-\pi_{\mu,0}(f)\|_{op}=0.$**

Proof.

Let $f\in C_{0,2}(G_{0})$

1. (1)

We have the following bound for the kernel functions $f_{\mu,H}$ where $(\mu,H)\in\Gamma_{0}\cup\Gamma_{1}$

[TABLE]

Since the dimension of the representation $\rho_{\mu}$ of $K_{H}$ is finite (denote by $d_{\mu}$ ), it follows that the norme $\|\cdot\|_{\rm op}$ and $\|\cdot\|_{H.S}$ on the space ${\mathcal{H}}_{\rho_{\mu}}$ are equivalent and

[TABLE]

Then by $(\ref{compactoper})$ the operator $\pi_{(\mu,H)}(f)$ is Hilbert-Schmidt and its Hilbert-Schmidt norm is given by

[TABLE] 2. (2)

Let $(\mu,H_{n})_{n\in{\mathbb{N}}}$ be a sequence converges to $(\mu,H)$ in $\Gamma_{0}$. Using now $(\ref{expoppimuH})$, we obtain

[TABLE]

For $f\in C_{0,2}(G_{0})$ we have

[TABLE]

Hence

[TABLE]

Let now $(\mu,H_{n})_{n\in{\mathbb{N}}}$ be a sequence converges to $(\mu,H)$ in $\Gamma_{1}$. We have for $\xi\in L^{2}(K/K_{H},\mu)$

[TABLE]

since the function $(k,h,s,H)\longmapsto|\widehat{f}^{2}(hsk^{-1},Ad(h)H_{n})-\widehat{f}^{2}(hsk^{-1},Ad(h)H)|$ converges uniformly to [math] as $n$ tends to $\infty$, since

[TABLE]

we see therefore that

[TABLE]

uniformly in $\mu$. 3. (3)

It remains for us to prove that $\underset{|\mu|\to\infty}{\lim}\|\pi_{(\mu,H)}(f)\|_{H.S}=0.$ This is equivalent to show that

[TABLE]

Recall that

[TABLE]

Let

[TABLE]

So,

[TABLE]

Using now the Plancherel formula, we get

[TABLE]

Therefore

[TABLE] 4. (4)

From $(\ref{expoppimuH})$ we have for $\mu\in\widehat{K_{H}}$ and $\xi\in L^{2}(K/K_{H},\mu)$

[TABLE]

where

[TABLE]

Since the function $(k,h,s,H)\longrightarrow|\widehat{f}^{2}(hsk^{-1},Ad(h)H)-\widehat{f}^{2}(hsk^{-1},0)|$ converges uniformly to [math] as $H$ tends to [math] since

[TABLE]

we see therefore that

[TABLE]

uniformly in $\mu$.

The proposition follows now from the density of $C_{0,2}(G_{0})$ in $C^{*}(G_{0})$. ∎

3.2. A $C^{*}$-condition.

Definition 3.4**.**

For an operator field $F\in\ell^{\infty}(\widehat{G_{0}}),$ let

[TABLE]

We have

[TABLE]

Definition 3.5**.**

Let ${\mathcal{D}}$ be the familly consisting of all operator fields $F\in\ell^{\infty}(\widehat{G_{0}})$ satisfying the following conditions:

1. (1)

$F(\mu,H)$* is a compact operator on ${\mathcal{H}}_{\mu,H}$ for every $(\mu,H)\in\Gamma_{0}\sqcup\Gamma_{1}.$* 2. (2)

The mapping $\widehat{G_{0}}\longrightarrow{\mathcal{B}}({\mathcal{H}}_{\mu,H}):(\mu,H)\longmapsto F(\mu,H)$ are norm continuous on the difference sets $\Gamma_{i},\ i=0,\ 1,\ 2$. 3. (3)

$\underset{|\mu|\to\infty}{\lim}\|F(\mu,H)\|_{\rm op}=0.$** 4. (4)

$\underset{H\to 0}{\lim}\|F(\mu,H)-F(\mu,0)\|_{\rm op}=0$* uniformly in $\mu\in K_{H}.$* 5. (5)

$\underset{|\lambda|\to\infty}{\lim}\|F(\lambda)\|_{\rm op}=0.$**

Definition 3.6**.**

Let

[TABLE]

Remark 3.7**.**

According to Proposition $4.7$ in [1] the unitary dual $\widehat{{\mathcal{D}}_{0}}$ is in bijection with the parameter space $\Gamma_{0}\cup\Gamma_{1}.$

Theorem 3.8**.**

${\mathcal{D}}$* is a $C^{*}$-algebra for the norm $\|\cdot\|_{\rm op},$ which is isomorphic to the $C^{*}$-algebra of Cartan motion groups under the Fourier transform.*

Proof.

First we show that ${\mathcal{D}}$ is a $C^{*}$-algebra. It is clear that ${\mathcal{D}}$ is an involutive sub-algebra of $\ell^{\infty}(\widehat{G_{0}})$. The conditions $(1),\ (2),\ (3)$ et $(5)$ are evidently true for every $F\in{\mathcal{D}}$. For the condition $(4)$, let $(F_{k})_{k}\subset{\mathcal{D}}$ such that $\underset{k\to\infty}{\lim}\|F_{k}-F\|_{\infty}=0.$ We have then $\underset{H\to 0}{\lim}\|F_{k}(\mu,H)-F_{k}(\mu,0)\|_{\rm op}=0$ uniformly in $\mu$, since for any $\varepsilon>0$ there exists $k_{0}$ such that such that $\|F-F_{k}\|\infty<\varepsilon$ for any $k\geq k_{0}.$ Then $\|F_{k_{0}}(\mu,H)-F_{k_{0}}(\mu,0)\|_{\rm op}<\varepsilon$ uniformly in $\mu$. Hence

[TABLE]

Then $F\in{\mathcal{D}}$.

Second, we have the quotient space ${\mathcal{D}}/{\mathcal{D}}_{0}$ is isomorphic to $C^{*}(K)$, indeed, let

[TABLE]

We have $\hbox{ker}(\delta)={\mathcal{D}}_{0}.$ Since ${\mathcal{D}}$ contains the algebra ${\mathcal{F}}(C^{*}(G_{0}))$ the image of $\delta$ contains the image of the Fourier transform of $C^{*}(K)$. Then $\delta$ is surjective. In addition ${\mathcal{D}}$ is a type $I$ algebra and $\widehat{G_{0}}$ is in bijection with $\widehat{\mathcal{D}}$, indeed we have $\widehat{G_{0}}\subset\widehat{\mathcal{D}}$. Moreover for every for every irreducible representation $\pi\in\widehat{\mathcal{D}}$, we have either $\pi_{|{\mathcal{D}}_{0}}\neq\{0\}$, and then $\pi=\pi_{(\mu,H)}$ since ${\mathcal{D}}_{0}$ is closed ideal of ${\mathcal{D}}$ and $\widehat{{\mathcal{D}}_{0}}=\Gamma_{0}\bigcup\Gamma_{1}$ or $\pi\in\widehat{{\mathcal{D}}/{\mathcal{D}}_{0}}$ and $\pi\in\Gamma_{2}$. Hence $\pi\in\widehat{G_{0}}$ and $\widehat{G_{0}}=\widehat{\mathcal{D}}$ as sets.

Thanks to Stone-Weierstrass’s theorem we have the identity ${\mathcal{D}}={\mathcal{F}}(C^{*}(G_{0})).$ ∎