The C*-algebra of the semi-direct product K and A

TL;DR

This paper characterizes the C*-algebra of a semi-direct product group formed by a compact group acting on an abelian locally compact group, extending known results to a broader class of groups.

Contribution

It provides a general description of the C*-algebra of semi-direct product groups with compact and abelian components, generalizing earlier specific cases.

Findings

Describes the C*-algebra as an algebra of operator fields over the spectrum.

Generalizes previous results for special classes of semi-direct product groups.

Provides a framework for analyzing the structure of C*-algebras in this context.

Abstract

Let $G=K\ltimes A$ be the semi-direct product group of a compact group $K$ acting on an abelian locally compact group $A$. We describe the $C^*$-algebra $C^*(G)$ of $G$ in terms of an algebra of operator fields defined over the spectrum of $G $, generalizing previous results obtained for some special classes of such groups.

Equations

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

$$\mathcal{F}(c)(\pi )=\pi (c)\in \mathcal{B}({\mathcal{H}}_{\pi }),\text{ for all }\pi \in \hat{G},c\in C^{\ast }(G).$$

$${\ell }^{\infty }(\hat{G})=\left\{\varphi :\hat{G}⟶\bigcup _{\pi \in \hat{G}}\mathcal{B}({\mathcal{H}}_{\pi });\varphi (\pi )\in \mathcal{B}(\pi ),\Vert \varphi {\Vert }_{\infty }:=\underset{\pi \in \hat{G}}{\sup }\Vert f(\pi ){\Vert }_{\mathrm{op}}<\infty \right\}.$$

$$\widehat{C^{\ast }(K)}$$

$$\widehat{C^{\ast }(A)}$$

$$(k,a).(h,b)=(kh,h^{-1}\cdot a+b),\forall (k,a),(h,b)\in G,$$

$$k\cdot \chi (b)=\chi (k^{-1}\cdot b),b\in A.$$

$$K_{\chi }=\{k\in K|k\cdot \chi =\chi \}.$$

$$\delta (k,a)(f)(h)$$

$${\sigma }_{(\mu ,\chi )}:=\mu \otimes \chi$$

$$G_{\chi }:=K_{\chi }⋉A,$$

$${\tau }_{\lambda }(k,a):={\tau }_{\lambda }(k),(k,a)\in G.$$

$$\dim ({\mu }_{\mathrm{k}})={\mathrm{d}}_{{\mu }_{\mathrm{k}}}={\mathrm{d}}_{{\mu }^{\prime }}=\dim ({\mu }^{\prime })\text{ for all }\mathrm{k}\text{ in the subnet}.$$

$$L=\{{\pi }_{(\nu ,\chi )};\nu \in \widehat{K_{\chi }},{\nu }_{|K^{\prime }}\ni {\mu }^{\prime }\}.$$

$$L^2(K{)}^{\rho }$$

$${\xi }_j$$

$$c_{{\mu }^{\prime }}\ast {\mathcal{H}}_0^{\mu }$$

$$c_{i,j}^n(k)$$

$$c_{i,j}(k)$$

$$S_n$$

$${\Gamma }_n$$

$$S$$

$$L^2(G/K_{\chi }⋉A,{\sigma }_{(\mu ,\chi )})\simeq L^2(K/K_{\chi },\mu )\subset L^2(K,{\mathcal{H}}_{\mu }).$$

$${\pi }_{(\mu ,\chi )}(k,a)\xi (h)$$

$${\pi }_{(\mu ,\chi )}(f)\xi (h)$$

$$f_{\mu ,\chi }:K\times K$$

$$(h,k)$$

$${\hat{f}}^2(k,\mu )$$

$$\mathcal{F}(f)(\mu ,\chi )$$

$$L^1(G{)}_c$$

$$L^2(\nu )$$

Full text

1. Introduction

It is wellknown that for a simply connected nilpotent Lie group and more generally for an exponential solvable Lie group $G={\rm exp}(\mathfrak{g})$, its dual space $\widehat{G}$ is homeomorphic to the space of co-adjoint orbits $\mathfrak{g}^{*}/G$ through the Kirillov mapping (see [7]). If we consider semi-direct products $G=K\ltimes N$ of compact connected Lie groups $K$ acting on simply connected nilpotent Lie groups $N$, then again we have an orbit picture of the dual space of $G$ (see [8]) and one can imagine that the topology of $\widehat{G}$ is linked to the topology of the admissible co-adjoint orbits.

By definition the $C^{*}$-algebra $C^{*}(G)$ of a locally compact group $G$ is the completion of the convolution algebra $L^{1}(G)$ with respect to the norm

[TABLE]

The unitary dual or spectrum $\widehat{C^{*}(G)}$ of $C^{*}(G)$ is in bijection with the dual space $\widehat{G}$ of $G$. Define the Fourier transform ${\mathcal{F}}$ on $C^{*}(G)$ by

[TABLE]

Then $C^{*}(G)$ can be identified with the sub-algebra $\widehat{C^{*}(G)}$ of the big $C^{*}$-algebra $\ell^{\infty}(\widehat{G})$ of bounded operator fields given by

[TABLE]

Here ${\mathcal{B}}({\mathcal{H}})$ denotes the space of bounded linear operators on the Hilbert space ${\mathcal{H}}$.

In order to understand the structure of the algebra $C^{*}(G)$, we must determine the special conditions which determine the operator fields $\phi\in\widehat{C^{*}(G)}$.

For instance if $G=K$, then it is easy to see that

[TABLE]

Since $K$ is compact, its irreducible unitary representations are finite dimensional. So $\pi(c)$ is always trivially a compact operator. Furthermore, the spectrum of $K$ has the discrete topology. So there is no continuity condition for the operator fields.

If $G=A$ is abelian, then $\widehat{A}$ is a locally compact Hausdorff space and

[TABLE]

If $G$ is more generally a locally compact group of the form $K\ltimes A$, then its dual space $\widehat{G}$ has a complicated structure determined by the $K$-orbits in $\widehat{A}$ and the spectra of the stabilizer groups of the elements of $\widehat{A}$. The topology of $\widehat{G}$ has been described in the paper of W. Bagget [2], which will be used intensively here. For the motion groups of the form $G_{n}=SO(n)\ltimes{\mathbb{R}}^{n}$ the conditions for $\widehat{C^{*}(G_{n})}$ have been described explicitly in [3]. For some other groups only a detailed description of the topology of $\widehat{G}$ is known (see for instance [4] and [9]).

In section $2$, we recall the results of Bagget on the topology of $\widehat{K\ltimes A}$, define the Fourier transform for $C^{*}(G)$ and discover the conditions which determine the algebra $\widehat{C^{*}(G)}$ inside the big algebra $\ell^{\infty}(\widehat{G})$ (see Definition 2.20). To see what are the difficulties, consider a converging net $\Pi_{M}=(\pi_{m}=\pi_{\mu_{m},\chi_{m}})_{m\in M}$ of irreducible representations of $G$. Here $(\chi_{m})_{m\in M}$ is a net of unitary characters of the abelian group $A$, $K_{m}$ is the stabilizer of $\chi_{m}$ in $K$ and $\mu_{m}$ is an element of $\widehat{K_{m}},m\in M$. Then $\pi_{m}=\text{ind}_{K_{m}\ltimes A}^{G}\mu_{m}\otimes\chi_{m}$. The problem is to understand for $a\in C^{*}(G)$ the behavior of the net of operators $(\pi_{m}(a))_{m\in M}$ acting on the different Hilbert spaces ${\mathcal{H}}_{\pi_{m}}$. We shall in fact construct for a certain converging subnet of the net $\Pi_{M}$ a common Hilbert space ${\mathcal{H}}_{M}$, such that ${\mathcal{H}}_{M}$ contains a copy ${\mathcal{H}}_{m}^{\prime}$ of the Hilbert space ${\mathcal{H}}_{m}$ of $\pi_{m}$ and a projection $P_{m}:{\mathcal{H}}_{M}\to{\mathcal{H}}_{m}^{\prime}$ for every $m$ in the subnet and we show that the essential condition for an operator field $\Phi$ to be an element of $\widehat{C^{*}(G)}$ is an operator norm convergence of the net $(\Phi(\pi_{m}))_{m\in M}$ to a limit operator $\sigma(\Phi_{|L})\in{\mathcal{B}}({\mathcal{H}}_{M})$ which is determined by the restriction $\Phi_{|L}$ of the operator field $\Phi$ to the limit set $L$ of the subnet.

In the last paragraph we present as example the group $G_{n,m}=SO(n)\times SO(m)\ltimes{\mathbb{R}}^{n+m}$, for $n,m\in{\mathbb{N}}$.

2. The $C^{*}$-algebra of the group $G=K\ltimes A.$

2.1. Preliminaries.

Let $K$ be locally compact group, $A$ be an abelian and suppose $\Psi$ is homomorphism of $K$ into the group of automorphisms of $A$ such that the mapping \left.\begin{array}[]{cccc}&&&\\ \Psi:&K\times A&\longrightarrow&A\\ &(k,a)&\mapsto&\Psi(k)(a)\end{array}\right. is continuous. For simplicity, we write the action of the automorphism $\Psi(k)$ on an element $a$ of $A$ as $k\cdot a.$

The semi-direct product $G=K\ltimes A$ of the groups $K$ and $A$ is the following locally compact group. $G$ is the topological product of $K$ and $A$ and $G$ is equipped with the group law

[TABLE]

where we write the multiplication in $K$ with the symbol $\cdot$ and multiplication in $A$ additively as $+$.

The group $A$ can be homeomorphically and isomorphically identified with the closed normal subgroup of $G$ consisting of all pairs $(1_{K},a)$, where $a$ is an element of $A$. Also, $K$ is homeomorphic and isomorphic to the closed subgroup of $G$ consisting of all pairs $(k,0_{A})$, where $k$ is an element of $K$. (We write $1_{K}$ for the multiplicative identity of $K$, $0_{A}$ for the additive identity of $A$, and $(1_{K},0_{A})$ for the identity of $G$).

Now, let $k\in K$ and $\chi:A\to{\mathbb{T}}\in\widehat{A}$ be a unitary character of $A$, and let $k\cdot\chi$ to be the character of $A$ defined by

[TABLE]

Thus $K$ acts on the left as a group of continuous transformations on $\widehat{A}$. If $\chi$ is in $\widehat{A}$, the stability subgroup $K_{\chi}$ of $\chi$ is the closed subgroup

[TABLE]

Definition 2.1**.**

**

1. (1)

We define the space ${\mathcal{K}}(G)$ to be the collection of all closed subgroups of $G$ equipped with the compact-open topology $($see **[6]$)$. 2. (2)

Let ${\mathcal{A}}(G)$ denote the set of all pairs $(C,T)$ where $C$ is a closed subgroup of $G$ and $T$ is an irreducible unitary representation of $C.$

Proposition 2.2**.**

[see, [2] 2.1-D] Any closed subgroup of $G$ which contains $A$ is of the form $J\ltimes A$, where $J$ is a closed subgroup of $K$. Further, a net $(J_{n})_{n}$ of closed subgroups of $K$ converges to $J$ in ${\mathcal{K}}(K)$ if and only if the net $(J_{n}\ltimes A)_{n}$ of subgroups of $G$ converges to the subgroup $J\ltimes A$ in ${\mathcal{K}}(G).$

Let now $\chi\in\widehat{A}$. Then the Hilbert space of the representation $\delta=\text{ind}_{\{1_{K}\}\ltimes A}^{G}\chi$ can be identified with $L^{2}(K)$, thus if $(k,a)\in G$, $f\in L^{2}(K)$ and $h\in K$ we have

[TABLE]

2.2. The topology of the dual space of the group $G$.

The dual space or spectrum of $G$ has been described by G.Mackey (for details, see [10] and [11]).

For each character $\chi\in\widehat{A}$ and any irreducible unitary representation $\mu$ of the stabilizer $K_{\chi}$ of $\chi$ in $K$, we have that

[TABLE]

is an irreducible unitary representation of

[TABLE]

whose restriction to $A$ is a multiple of $\chi$ (see [2] proposition $2$ p.181) and the induced representation $\pi_{(\mu,\chi)}:=\text{ind}_{G_{\chi}}^{G}\sigma_{(\mu,\chi)}$ is an irreducible representation of $G$.

On the other hand, every irreducible unitary representation $\tau_{\lambda}$ of $K$ extends to an irreducible representation (also denoted by $\tau_{\lambda}$) of the entire group $G$ defined by

[TABLE]

The following Propositions give the relationship between $\widehat{G}$ and the set of all elements $\pi_{(\mu,\chi)}$ (see [12])

Proposition 2.3**.**

Let $\pi$ be an irreducible unitary representation of $G$. Then $\pi_{|A}$ is supported by a $K$-orbit $\theta$ in $\widehat{A}$. Suppose $\chi$ is an element of $\theta$. Then $\pi$ is equivalent to a representation of the form $\pi_{(\mu,\chi)}$ where $\mu$ is an irreducible unitary representation of $K_{\chi}.$

Proposition 2.4**.**

Let $\pi$ and $\theta$ be as in the above proposition. Assume $\chi$ and $\chi^{\prime}$ are elements of $\theta.$ i.e, $\chi^{\prime}=k\cdot\chi$ for some element $k$ of $K$. Suppose further that $\pi$ is equivalent to $\pi_{(\mu,\chi)}$ and also is equivalent to $\pi_{(\mu^{\prime},\chi^{\prime})}$ for two elements $\mu\in\widehat{K_{\chi}},\mu^{\prime}\in\widehat{K}_{\chi^{\prime}}$. Then

1. (1)

$G_{\chi^{\prime}}=k\cdot G_{\chi}\cdot k^{-1}.$** 2. (2)

the representation $h\mapsto\mu^{\prime}(khk^{-1}),h\in K_{\chi}$, is equivalent to the representation $\mu$.

Definition 2.5**.**

A cataloguing triple we mean a triple $(\chi,J,\mu)$, where $\chi$ is a character of $A,$ $J$ is the stabilizer $K_{\chi}$ and $\mu$ is an irreducible unitary representation of $K_{\chi}.$ We denote by $\pi_{\chi,J,\mu}:=\pi_{(\mu,\chi)}$ the induced representation $\displaystyle\text{ind}_{{}_{J\ltimes A}}^{G}(\mu\otimes\chi).$

By Baggett in [2] (Proposition $2.4$-$D$ $p.187$), we have

Proposition 2.6**.**

The mapping $(\chi,K_{\chi},\mu)\longrightarrow\pi_{(\mu,\chi)}$ is onto $\widehat{G}.$

For the proof of the following theorem, see $\S\S 4.\ 1$-$D$ of [2].

Theorem 2.7**.**

The $C^{*}$-algebra $C^{*}(G)$ of the locally compact group $G=K\ltimes A$ is $CCR$.

Definition 2.8**.**

Let $F$ be the set of all functions $f$ which satisfy:*

1. (1)

The domain of $f$ is a closed subgroup of $G$. 2. (2)

$f$* is complex-valued and continuous on its domain.*

In the section $3$ of [6], Fell has defined a topology on $F$.

Definition 2.9**.**

The Fell topology on $F$ is the topology defined as follows. Let $(f_{n})$ be a net of elements of $F$. For each $n$, let $H_{n}$ be the domain of $f_{n}$. Let $f$ be in $F$ and denote by $H$ the domain of $f$. Then the net $f_{n}$ converges to $f$ in the Fell topology if and only if the following two conditions hold.

1. (1)

The net $H_{n}$ converges to $H$ in ${\mathcal{K}}(G).$ 2. (2)

For each subnet $(f_{n_{m}})_{m}$ of the net $(f_{n})_{n}$ and for each net $h_{m}$ of points of $G$ such that for each $m,$ $h_{m}$ is an element of $H_{n_{m}}$ and such that the net $(h_{m})_{m}$ converge to an element $h$ of $H$, the net of complex numbers $(f_{n_{m}}(h_{m}))_{m}$ converges to $f(h),$

In [6] Fell describes the topology on ${\mathcal{A}}(G)$ in terms of the elements of $F$ and in terms of the Fell topology on $F$. Here is one consequence of that description.

Theorem 2.10**.**

Let $(H_{n},S_{n})_{n}$ be a net of elements of ${\mathcal{A}}(G)$ and $(H,S)$ be an element of ${\mathcal{A}}(G)$. Assume that, for some function of positive type $f$ associated with $S$, there exists a net $(f_{n})_{n}$ of functions which satisfies:

1. (1)

Each $f_{n}$ is a finite sum of functions of positive type associated with $S_{n}$. 2. (2)

The net $(f_{n})_{n}$ converges to $f$ in the Fell topology of $F$.

Then the net $(H_{n},S_{n})_{n}$ converges to $(H,S)$ in ${\mathcal{A}}(G).$

The topology of the dual space of the group $G$ has been described by Baggett in [2] in the following theorem:

Theorem 2.11**.**

The topology of $\widehat{G}$ may be described as follows: Let $B$ be a subset of $\widehat{G}$ and $\pi$ an element of $\widehat{G}$. $\pi$ is contained in the closure of $B$ if and only if there exist: a cataloguing triple $(\chi,(K_{\chi},\mu))$ for $\pi$, an element $(H,S)$ of ${\mathcal{A}}(K)$ and a net $(\chi_{n},(K_{\chi_{n}},\mu_{n}))_{n}$ of cataloguing triples, such that:

1. (1)

For each $n$, the element $\pi_{(\mu_{n},\chi_{n})}$ of $\widehat{G}$ is an element of $B.$ 2. (2)

The net $(\chi_{n},(K_{\chi_{n}},\mu_{n}))$ converges to $(\chi,(K^{\prime},\mu^{\prime}))$ in $\widehat{A}\times{\mathcal{A}}(K).$ 3. (3)

$K_{\chi}$* contains $K^{\prime}$, and $\text{ind}_{K^{\prime}}^{K_{\chi}}\mu^{\prime}$ contains $\mu$.*

The following lemma is the key for Definition 2.20.

Lemma 2.12**.**

Let $(\chi_{n},(K_{\chi_{n}},\mu_{n}))_{n}$ be a properly converging net with the limit $(\chi,(K^{\prime},\mu^{\prime}))$ (i.e $K^{\prime}$ is a closed subgroup of $K_{\chi}$ and $\mu^{\prime}\in\widehat{K^{\prime}}$). Then for some subnet we have that

[TABLE]

Proof.

The limit set $L$ of the net $(\pi_{k})_{k}$ in $\widehat{G}$ is according to [2] the set

[TABLE]

We can as in [2] realize all these representations $\pi_{(\mu,\chi)}$ on subspaces in the common Hilbert space $L^{2}(K)$. Take some $\mu\in\widehat{K_{\chi}}$ such that $\mu_{|K^{\prime}}$ contains $\mu^{\prime}$. Choose a $\rho\in\widehat{K}$, such that $\rho_{|K_{\chi}}$ contains $\mu$. Let $L^{2}(K)^{\rho}$ be the minimal left and right translation $K$-invariant subspace of $L^{2}(K)$ containing ${\mathcal{H}}_{\rho}$ and thus also a copy of the Hilbert space ${\mathcal{H}}_{\mu}$ of the representation $\mu$ of $K_{\chi}$. Then $L^{2}(K)^{\rho}$ has dimension $d_{\rho}^{2}$.

Then $L^{2}(K)^{\rho}$ also contains a copy $({\mathcal{H}}_{\mu^{\prime}},\mu^{\prime})$ of the irreducible representation $\mu^{\prime}$ of $K^{\prime}$. Let ${\mathcal{H}}^{\mu^{\prime}}$ be the $\mu^{\prime}$-isotopic component inside ${{\mathcal{H}}^{\rho}}$. We write

[TABLE]

where ${\mathcal{H}}^{\rho}_{0}:=({\mathcal{H}}^{\mu^{\prime}})^{\perp}\subset L^{2}(K)^{\rho}$. There exists $l_{\mu}\in{\mathbb{N}}_{>0}$ such that ${\mathcal{H}}^{\mu^{\prime}}\simeq l_{\mu}{\mathcal{H}}_{\mu^{\prime}}$. Similarly we have $L^{2}(K)^{\rho}={\mathcal{H}}^{\mu_{n}}\oplus{\mathcal{H}}^{\mu_{n}}_{0}$, $\mu_{|K_{\chi_{n}}}\simeq l_{n}\mu_{n}$, ${\mathcal{H}}^{\mu_{n}}\simeq l_{n}{\mathcal{H}}_{\mu_{n}}$ for some $l_{n}\in{\mathbb{N}}$. Of course $l_{n}\leq\rm{dim}(L^{2}(K)^{\mu})$ and $l_{n}$ can be [math].

However, since the representations $(\pi_{(\mu_{n},\chi_{n})})$ converge to the representation $\pi_{(\mu,\chi)}$, we can assume that for a subnet the subspaces ${\mathcal{H}}^{\mu_{n}}$ can be realized inside $L^{2}(K)^{\rho}$ for $n$ large enough (see [2], 4.2-D Theorem). This tells us also that $l_{n}>0$ for $n$ large enough and hence, again for a subnet, we can suppose that $l_{n}=l$ is fixed for every $n$. Since the dimensions of the spaces ${\mathcal{H}}^{\mu_{n}}$ are smaller than the dimension of $L^{2}(K)^{\rho}$, we can also assume that all the dimensions $d_{\mu_{n}}$ are the same and equal to some common $d>0$.

We choose for every $n$ an orhonormal basis $(\xi^{n}_{j})_{j}$ of $L^{2}(K)^{\rho}$, which passes through the $l_{n}$ copies of ${\mathcal{H}}^{\mu_{n}}$ and through ${\mathcal{H}}^{\mu_{n}}_{0}$. We choose the $\xi^{n}_{j}$ such that $\text{span}\{\xi^{n}_{1},\cdots,\xi^{n}_{d_{\mu_{n}}}\}$ is a copy of the space ${\mathcal{H}}_{\mu_{n}}$. Since the dimension of $L^{2}(K)^{\rho}$ is finite, we can assume (passing to a subnet) that $\lim_{n\to\infty}\xi_{j}^{n}=\xi_{j}$ exists in $L^{2}(K)$ for every $j\leq ld$. Let $c_{n},n\in N,$ be the character of the irreducible representation $\mu_{n}$ of $K^{\prime}$. Then $c_{n}\ast\xi_{n}=\xi_{n}$ for any $\xi_{n}\in{\mathcal{H}}^{\mu_{n}}$ and $c_{n}\ast H^{\mu_{n}}_{0}=\{0\}$ for every $n$. According to Bagget, the pairs $(K_{\chi_{n}},c_{n})$ converge in ${\mathcal{A}}(K)$ to the pair $(K^{\prime},c_{\mu^{\prime}})$, where $c_{\nu}$ denotes the character of an irreducible representation $\nu$ (see [2], 7.1-B Lemma). This implies that (see [2],1.4-A Proposition)

[TABLE]

for every $j$ and similarly

[TABLE]

This shows that $K^{\prime}$ acts on ${\mathcal{H}}^{\mu^{\prime}}=\lim_{n\to\infty}{\mathcal{H}}^{\mu_{n}}$ by a multiple of $\mu^{\prime}$. Define for any $n$ and $i,j\in\{1,\cdots,d\}$ the function $c^{n}_{i,j}$ on $K$ by

[TABLE]

where $\lambda$ denotes the left regular representation of $K$ on $L^{2}(K)$. Since $\lim_{n\to\infty}\xi^{n}_{i}=\xi_{i}$ in $L^{2}(K)$ for every $i$, the functions $c^{n}_{i,j}$ converge uniformly on $K$ to the function $c_{i,j}$, where

[TABLE]

Now the operator ${\mu_{n}(c^{n}_{i,j}}_{|K_{\chi_{n}}})$, which acts on the Hilbert space ${\mathcal{H}}_{\mu_{n}}:=\text{span}\{\xi^{n}_{j},1\leq j\leq d,\}$ is the rank one operator $P_{\xi^{n}_{i},\xi^{n}_{j}}$ which converges to the operator $P_{\xi_{i},\xi_{j}}$ on the Hilbert space ${\mathcal{H}}_{\mu^{\prime}}:=\lim_{n\to\infty}{\mathcal{H}}_{\mu_{n}}$. This shows that the restriction of $\mu^{\prime}$ to ${\mathcal{H}}_{\mu^{\prime}}$ is irreducible. Hence $d=\rm{dim}({\mathcal{H}}_{\mu^{\prime}})$ and $l_{\mu^{\prime}}=l$. ∎

2.3.

Remark 2.13**.**

Identifying for every $n$ the Hilbert space ${\mathcal{H}}_{\mu_{n}}$ with ${\mathbb{C}}^{d}$ via the basis given by the $\xi_{j}$’s, we see also that for every $f\in C(K)$ the operators $\mu_{n}(f_{|K_{\chi_{n}}})$ converge strongly and hence in operator norm to the operator $\mu^{\prime}(f_{|K^{\prime}})$.

We have by $\S\S 4.5$ of [5]:

Theorem 2.14**.**

Let $A$ be a postliminal $C^{*}$-algebra. Then $A$ admits a composition net $(I_{n})_{0\leq n\leq\alpha}$ such that for any $n$, which is not an ordinal, the quotient $I_{n+1}/I_{n}$ is with continuous trace and such that for every ordinal $\beta\leq\alpha$ the relation $I_{\beta}=\bigcup_{n<\beta}I_{n}$ holds.

We take now as $C^{*}$-algebra our $A=C^{*}(G)$, which is $CCR$.

Let

[TABLE]

Then $S_{0}=\widehat{G}$ and $S_{\alpha}=\{\emptyset\}$. The subsets

[TABLE]

are locally compact and Hausdorff in their relative topologies, since $\Gamma_{n}$ is the spectrum of the algebra $I_{n+1}/I_{n}$, which is of continuous trace (see [5]). Let $S=\widehat{G}$ be the spectrum of $G$. Then

[TABLE]

2.4. The Fourier transform.

Let us first write down explicitly the representation $\pi_{(\mu,\chi)}$. Its Hilbert space ${\mathcal{H}}_{{(\mu,\chi)}}$ can be identified with the space

[TABLE]

Let $\xi$ be an element of ${\mathcal{H}}_{{(\mu,\chi)}}.$ For all $a\in A$ and $k,h\in K$ we use the same calculation as in (2.1) and we have that

[TABLE]

Let us compute for $f\in L^{1}(G)$ the operator $\pi_{(\mu,\chi)}(f).$ We have for $h\in K$ and $\xi\in{\mathcal{H}}_{{(\mu,\chi)}}$ that

[TABLE]

where

[TABLE]

and

[TABLE]

Definition 2.15**.**

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

[TABLE]

Let now $L^{1}(G)_{c}$ be the dense subspace of $L^{1}(G)$ defined by

[TABLE]

Definition 2.16**.**

Let $L$ be a closed subgroup of the compact group $K$ and let $(\nu,{\mathcal{H}}_{\nu})$ be an irreducible representation of $L$ with character $\chi_{\nu}$. We may identify the Hilbert space ${\mathcal{H}}_{\nu}$ with ${\mathbb{C}}^{d}$, $d=d_{\nu}=\rm{dim}(\nu)$. Let*

[TABLE]

1. (1)

Define for $\chi\in\widehat{A}$ such that $L\subset K_{\chi}$ and for $f\in L^{1}(G)$ the operator $\tau_{\nu,\chi}(f)$ on $L^{2}(K,{\mathbb{C}}^{d})$ by

[TABLE] 2. (2)

Define for $\chi\in\widehat{A}$, for a closed subgroup $L\subset K_{\chi}$ of $K$, for $\nu\in\widehat{L}$, the linear projection $P_{\nu}:L^{2}(K,{\mathcal{H}}_{\nu})\to L^{2}(K/L,\nu)$ by

[TABLE]

Proposition 2.17**.**

**

1. (1)

The linear operator $P_{\nu}$ is an selfadjoint projection of the Hilbert space $L^{2}(\nu)$. 2. (2)

For any closed subgroup $L\subset K_{\chi}$ of $K$, $\nu\in\widehat{L}$, $\chi\in\widehat{A}$ and $f\in L^{1}(G)$ we have

[TABLE]

Proof.

1. (1)

We have for $\varphi\in L^{2}(K,{\mathcal{H}}_{\nu})$ that

[TABLE]

Let $\varphi\in{\mathcal{H}}_{\nu,\chi}=L^{2}(K/L,\nu)\subset L^{2}(K,{\mathcal{H}}_{\nu})$. Then for $k\in K$,

[TABLE]

Hence the operator $P_{\nu}$ is the identity on ${\mathcal{H}}_{\nu}\subset L^{2}(K,{\mathcal{H}}_{\nu})$.

Let $\mu\in\widehat{L}$ and let $c_{\mu}$ its character. For $\varphi\in L^{2}(K,{\mathcal{H}}_{\nu})$ let

[TABLE]

Then the mapping $\varphi_{\mu}$ is also contained in $L^{2}(\nu)$ and for another $\omega\in\widehat{L}$ we have that

[TABLE]

It is easy to see now that

[TABLE]

since $L^{2}(K)=\sum_{\mu\in\hat{L}}L^{2}(K)\ast c_{\mu}$. Furthermore, for $\mu\neq\nu$ it follows that

[TABLE]

Hence $L^{2}(\nu)^{\perp}=\sum_{\mu\neq\nu}L^{2}(\nu)_{\mu}$ and $P_{\nu}$ is zero on $L^{2}(\nu)^{\perp}$. This shows that

[TABLE] 2. (2)

For $f\in C_{c}(G),\varphi\in L^{2}(K,{\mathbb{C}}^{d}),x\in K$, we have by (2.2) that

[TABLE]

[TABLE]

∎

Lemma 2.18**.**

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

1. (1)

$\underset{(\mu,\chi)\longrightarrow\infty}{\lim}\|\mathcal{F}(f)(\mu,\chi)\|_{op}=0$* .* 2. (2)

Let $(\chi_{m},L_{m},\mu_{m})_{m\in M}$ be a converging net in $\widehat{A}\times{\mathcal{A}}(K)$ with limit $(\chi_{\infty},L_{\infty},\mu_{\infty})$ such that $\rm{dim}(\mu_{m})=\rm{dim}(\mu_{\infty})$ for any $m$. Then, identifying the Hilbert spaces of the representations $\mu_{m}$ with ${\mathbb{C}}^{d}$, we have that $\lim_{m\to\infty}\tau_{\mu_{m},\chi_{m}}(f)=\tau_{\mu_{\infty},\chi_{\infty}}(f)$ in operator norm for any $f\in L^{1}(G)$.

Proof.

1. (1)

Let $A$ be a $C^{*}$-algebra. According to ([5], chapter 3 §3.3) , if a net $(\pi_{k})_{k}\subset\widehat{A}$ goes to infinity, i.e. this net has no converging subnet, then $\lim_{k\to\infty}\|\pi_{k}(a)\|_{\rm op}=0$ for any $a\in A.$ Now Bagget [2] has shown that for every net of cataloguing triples $(\chi_{k},K_{\chi_{k}},\mu_{k})_{k}$ we have that $(\mu_{k},\chi_{k})_{k}$ goes to infinity, if and only if the net $(\pi_{(\mu_{k},\chi_{k})})_{k}$ goes to infinity in $\widehat{C^{*}(G)}$. 2. (2)

Let first $f$ be contained in $L^{1}(G)_{c}$. Let

[TABLE]

Then we see that $f_{m}(x,y)\in{\mathcal{B}}({\mathbb{C}}^{n})$ and by Remark that 2.13

[TABLE]

point wise in $x,y$. Therefore also

[TABLE]

where $M_{d}({\mathbb{C}})$ denotes the space of complex matrices of size $d$.

Using Lebesgue, we see that

[TABLE]

Hence

[TABLE]

The lemma follows now from the density of $L^{1}(G)_{c}$ in $C^{*}(G)$.

∎

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

Let $G=K\ltimes A$ be as before a semi-direct product of a compact group $K$ with a locally compact abelian group $A$.

Remark 2.19**.**

Let $(\pi_{(\mu_{m},\chi_{m})})_{m\in M}$ be a net in $\widehat{G}$ which converges to $\pi_{(\mu_{\infty},\chi_{\infty})}$. We can suppose that for a subnet (also denoted by $M$ for simplicity of notation) that the triples $(\chi_{m},K_{\chi_{m}},\mu_{m})$ converge to $(\chi_{\infty},K_{\infty},\mu_{\infty})$ in $\widehat{A}\times{\mathcal{A}}(K)$ and that the Hilbert spaces ${\mathcal{H}}_{\mu_{m}}$ and ${\mathcal{H}}_{\mu_{\infty}}$ are identified with ${\mathbb{C}}^{d}$ for some $d\in{\mathbb{N}}^{*}$ and that all these spaces ${\mathcal{H}}_{(\mu_{m},\chi_{m})},m\in M\cup\{\infty\}$, are subspaces of the common Hilbert space ${\mathcal{H}}_{M}:=L^{2}(K,{\mathbb{C}}^{d})$. The representation $\tau_{\mu_{\infty},\chi_{\infty}}=\text{ind}_{K_{\infty}\times A}^{G}\mu_{\infty}\otimes\chi_{\infty}$ can be disintegrated into an integral of irreducible representations supported by the limit set

[TABLE]

of the net $(\pi_{(\mu_{m},\chi_{m})})_{m}$ (see Theorem 2.11). We denote by $\sigma_{\mu_{\infty},\chi_{\infty}}$ the corresponding representation of the algebra $\widehat{C^{*}(G)}_{|L}$ on the Hilbert space $L^{2}(K/K_{\infty},\mu_{\infty})\subset L^{2}(K,{\mathbb{C}}^{d})$. Let us observe that by the construction of $\sigma_{\mu_{\infty},\chi_{\infty}}$ we have that

[TABLE]

We can extend this representation $\sigma_{\mu_{\infty},\chi_{\infty}}$ to the larger $C^{*}$-algebra $CB(L)$ consisting of all uniformly bounded operator fields $F$ satisfying $F(\pi)\in{\mathcal{K}}({\mathcal{H}}_{\pi}),\pi\in L,$ and we denote this extension also by $\sigma_{\mu_{\infty},\chi_{\infty}}$ (see [1]) .

Definition 2.20**.**

Let ${\mathcal{D}}(G)={\mathcal{D}}$ be the family consisting of all uniformly bounded operator fields $F\in\ell^{\infty}(\widehat{G})$ satisfying the following conditions:

1. (1)

$F(\pi)$* is a compact operator on ${\mathcal{H}}_{\pi}$ for every $\pi\in\widehat{G}.$* 2. (2)

$\underset{(\mu,\chi)\to\infty}{\lim}\|F(\mu,\chi)\|_{\rm op}=0.$** 3. (3)

Let $(\pi_{(\mu_{m},\chi_{m})})_{m\in M}$ be a properly converging net in $\widehat{G}$ with the properties and notations of the preceding Remark 2.19. Then

[TABLE]

Proposition 2.21**.**

${\mathcal{D}}(G)$* is a $C^{*}$-algebra for the norm $\|\cdot\|_{\rm op}$ containing $\widehat{C^{*}(G)}$.*

Proof.

First we show that ${\mathcal{D}}$ is a norm closed involutive subspace of $l^{\infty}(\widehat{G})$. It is clear that ${\mathcal{D}}$ is a sub-space of $\ell^{\infty}(\widehat{G})$. The conditions $(1),\ (2)$ are evidently true for every $F$ in the closure $\overline{{\mathcal{D}}}$ of ${\mathcal{D}}$. For the condition $(3)$, let $F\in\overline{{\mathcal{D}}}$ and let $(F^{k})_{k}\subset{\mathcal{D}}$ such that $\underset{k\to\infty}{\lim}\|F^{k}-F\|_{\infty}=0.$ Then also $\lim_{k\to\infty}\|(F^{k})^{*}-F^{*}\|_{\infty}=0$. Hence for any $\varepsilon>0$ there exists $k_{0}$ such that such that $\|F-F^{k}\|_{\infty}<\varepsilon$ for any $k\geq k_{0}.$ Therefore choosing some $k>k_{0}$ we have for $m$ large enough that

[TABLE]

and so

[TABLE]

Hence $F\in{\mathcal{D}}$. Since $\sigma_{\mu_{\infty},\chi_{\infty}}$ is a representation, it follows that ${\mathcal{D}}$ is involutive and so ${\mathcal{D}}$ is an involutive Banach space. Let us show that it is an algebra.

Let $F,F^{\prime}\in{\mathcal{D}}$. We must show that $F\circ F^{\prime}$ is in ${\mathcal{D}}$ too.

The conditions $(1),(2)$ are necessarily true for $F\circ F^{\prime}$.

Let us check point (3). It follows from property $(3)$ for $F$ , using the involution ∗, that also

[TABLE]

We then have that, since $P_{\mu}\circ F(\mu,\chi)\circ P_{\mu}=F(\mu,\chi)\circ P_{\mu}$ for every $\chi\in\widehat{A},\mu\in\widehat{K_{\chi}}$,

[TABLE]

where $C=\max(\|F\|_{\infty},\|F^{\prime}\|_{\infty})$.

Since $\widehat{C^{*}(G)}$ satisfies all the conditions of $D^{*}(G)$ it follows that $\widehat{C^{*}(G)}$ is contained in $D^{*}(G)$. ∎

Proposition 2.22**.**

The spectrum $\widehat{{\mathcal{D}}(G)}$ of the algebra ${\mathcal{D}}(G)$ can be identified with $\widehat{G}.$

Proof.

We have by $\S\S 4.5$ of [5]:

Theorem 2.23**.**

Let $A$ be a postliminal $C^{*}$-algebra. Then $A$ admits a composition sequence $(I_{n})_{0\leq n\leq\alpha}$ such that the quotients $I_{n+1}/I_{n}$ are $C^{*}$- algebras with continuous trace.

This theorem applies of course to our $C^{*}$-algebra $C^{*}(G)$. Let now

[TABLE]

The subsets

[TABLE]

are locally compact and Hausdorff in their relative topologies, since $\Gamma_{n}$ is the spectrum of the algebra $I_{n+1}/I_{n}$, which is of continuous trace (see [5]). Then

[TABLE]

Evidently $\widehat{{\mathcal{D}}}\supset\widehat{G}$. Define:

[TABLE]

Then the $J_{n}$’s are closed ideals of ${\mathcal{D}}$ and

[TABLE]

Let now $\pi\in\widehat{\mathcal{D}}\setminus\widehat{G}$. Let

[TABLE]

If $n_{\pi}=\alpha$, then $\pi(J_{n})=\{0\}$ for every $0\leq n<\alpha$ and then $\pi({\mathcal{D}})=\pi(J_{0})=\{0\}$, which is impossible. Hence

[TABLE]

We have now that $\pi(J_{n_{\pi}+1})\neq\{0\}$, but $\pi(J_{n_{\pi}})=\{0\}$.

This means in particular that $\pi$ is contained in the hull of the ideal $J_{n_{\pi}}$. But $J_{n_{\pi}}$ is the kernel of the subset $S_{n_{\pi}}$. In other words $\pi$ is an element of the closure of the subset ${S_{n_{\pi}}}$, but $\pi\not\in\overline{S_{{n_{\pi}+1}}}$.

There exists therefore a net $(\pi_{k})_{k}\subset\Gamma_{n_{\pi}}$, such that $\pi=\lim_{k\to\infty}\pi_{k}$ in $\widehat{\mathcal{D}}$. Hence, there exists for $\xi\in{\mathcal{H}}_{\pi}$ and for any $k$ an element $\xi_{k}\in{\mathcal{H}}_{\pi_{k}}$, such that for any $F\in{\mathcal{D}}$, we have that

[TABLE]

Then $\pi_{k}$ does not go to infinity in $\widehat{G}$ because of condition $(2)$. Hence, either for a subnet, the net $(\pi_{k}=\pi_{(\mu_{k},\chi_{k})})_{k}$ converges in $\Gamma_{n_{\pi}}$ to some $\pi_{\infty}=\pi_{(\mu_{\infty},\chi_{\infty})}\in\Gamma_{n_{\pi}}$, and then by condition (3)

[TABLE]

for any $F\in{\mathcal{D}}$. Now for any $k$ we have that

[TABLE]

This shows that

[TABLE]

Since in our case $L=\{\pi_{\infty}\}$, relation 2.4 implies that $\hbox{ker}(\pi_{\infty})\subset\hbox{ker}(\pi)$. But then $\pi=\pi_{\infty}$, since $\pi_{\infty}$ is completely continuous (see [5], 4.1.11. Corollary).

The other possibility is that the net converges in $\widehat{C^{*}(G)}$ to a limit set $L$ contained in $S_{n_{\pi}+1}$.

Now for $F\in J_{n_{\pi}+1}$, it follows from condition $(3)$, that

[TABLE]

which shows that $\lim_{k\to\infty}\|F(\pi_{k})\|_{\rm op}=0$. But this implies then by (2.5) that $\pi(F)(\xi)=0$ for every $\xi$. Therefore $\pi$ is [math] on $J_{n_{\pi}+1}$. This contradiction shows that $\pi=\pi_{\infty}\in\widehat{G}$. ∎

Theorem 2.24**.**

Let $G=K\ltimes A$ be the semi-direct product of a compact group $K$ with an abelian locally compact group $A$. Then the $C^{*}$-algebra ${\mathcal{D}}(G)$ is isomorphic to the group $C^{*}$-algebra $C^{*}(G)$.

Proof.

We know now that $\widehat{{\mathcal{D}}}=\widehat{G}$. It suffices to apply the theorem of Stone-Weierstrass to the $C^{*}$-algebra ${\mathcal{D}}$ and its subalgebra $\widehat{C^{*}(G)}$.

∎

3. Examples

Example 3.1**.**

For all $n\in{\mathbb{N}}^{*}$, let $G_{n}$ the the the semi-direct product of the compact Lie group $SO(n)$ with the abelian group ${\mathbb{R}}^{n}$. The $C^{*}$-algebra of this group is discribed by Abdelmoula, Elloumi and Ludwig in [3].

We can parameterize the dual space in the following way:

[TABLE]

For $\rho_{\mu}\in\widehat{SO(n-1)},$ and $r\in{\mathbb{R}}_{+}^{*}$, we denote by $K_{\chi_{r}}=SO(n-1)\times{\mathbb{R}}_{+}^{*}$ the stabilizer of $\chi_{r}$. The projection $P_{r}:L^{2}(SO(n))\longrightarrow L^{2}(SO(n))$ are

[TABLE]

and for $r=0$

[TABLE]

For any $f\in L^{1}(G_{n})$ and $\xi\in L^{2}(SO(n))$ we have that

[TABLE]

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

1. (1)

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

$\underset{\gamma\to\infty}{\lim}\|F(\gamma)\|_{\rm op}=0,$ 3. (3)

[TABLE]

Then, the $C^{*}$-algebra of the group the group $G_{n}$ is isomorphic to ${\mathcal{D}}_{n}$ under the Fourier transform.

Example 3.2**.**

Define the abelian group $A$ and the compact groups $L$ and $K$ by:

[TABLE]

Then, by [13], Ch. 4, 2, 3.1., we have that

[TABLE]

Furthermore, the spectrum $\widehat{K}$ of the abelian group $K$ is the set of all infinite products

[TABLE]

where $\chi_{j}=1$ almost everywhere.

Define for $m\in{\mathbb{N}}$ the subgroups $L^{m}$ and $L_{m}$ of $K$ by

[TABLE]

and

[TABLE]

Then $L^{m}$ is isomorphic to the quotient group $K/L_{m}$ and it has $2^{m}$ elements. The Haar measure $dl^{m}$ on $L^{m}$ is given by

[TABLE]

A function $f:K\to{\mathbb{C}}$ is continuous, if and only if for every $\varepsilon>0$, there exists $m\in{\mathbb{N}}$ such that

[TABLE]

Then the Haar measure $dk$ on $K$ is given according to [13], Ch. 3, 3, example (vi) by

[TABLE]

For $\chi=(\chi_{i})_{i\in{\mathbb{N}}}\in\widehat{A}$, the stabilizer $K_{\chi}$ is the direct product

[TABLE]

where $K_{\chi_{i}}=\{1,-1\}$ if $\chi_{i}\in\{1,-1\}$ and $K_{\chi_{i}}=\{1\}$ if $\chi_{i}\not\in\{1,-1\}$.

Let for $m\in{\mathbb{N}}$

[TABLE]

Then

[TABLE]

in $\widehat{A}$. The stabilizer $K_{\chi_{m}}$ in $K$ is the subgroup

[TABLE]

Then

[TABLE]

Choose for every $m\in{\mathbb{N}}$ the trivial character $\mu_{m}$ of $K_{\chi_{m}}$. Then, according to Definition 2.16, the dimensions $d_{\mu_{m}}$ are one and the projections $P_{\mu_{m}}:L^{2}(K)\to L^{2}(K)$ are given by

[TABLE]

Therefore, by Remark 2.19, the limit set of the sequence $(\pi_{\mu_{m},\chi_{m}})$ is the spectrum $\widehat{K}$ itself and then for any $f\in L^{1}(G)$ and $\xi\in L^{2}(K)$ we have that

[TABLE]