Weak containment by restrictions of induced representations

TL;DR

This paper investigates the properties of QSIN groups, showing how restrictions of induced representations relate to weak containment, and explores implications for the structure of their C*-algebras.

Contribution

It establishes a key weak containment result for representations of QSIN groups and analyzes the consequences for their C*-algebra properties.

Findings

If G is a QSIN group with a copy of F_2, then C*(G) is not locally reflexive.

C*_r(G) does not have the local lifting property.

Provides tools for studying Fourier and Fourier-Stieltjes spaces.

Abstract

A QSIN group is a locally compact group $G$ whose group algebra $L^1(G)$ admits a quasi-central bounded approximate identity. Examples of QSIN groups include every amenable group and every discrete group. It is shown that if $G$ is a QSIN group, $H$ is a closed subgroup of $G$, and $\pi$ is a unitary representation of $H$, then $\pi$ is weakly contained in $(\mathrm{Ind}_H^G\pi)|_H$. This provides a powerful tool in studying the C*-algebras of QSIN groups. In particular, it is shown that if $G$ is a QSIN group which contains a copy of $\mathbb F_2$ as a closed subgroup, then $C^*(G)$ is not locally reflexive and $C^*_r(G)$ does not admit the local lifting property. Further applications are drawn to the "(weak) extendability" of Fourier spaces $A_\pi$ and Fourier-Stieltjes spaces $B_\pi$.

Equations

$$0\to {\mathrm{C}}^{\ast }({\mathbb{F}}_2){\otimes }_{\min }J\to {\mathrm{C}}^{\ast }({\mathbb{F}}_2){\otimes }_{\min }{\mathrm{C}}^{\ast }({\mathbb{F}}_2)\to {\mathrm{C}}^{\ast }({\mathbb{F}}_2){\otimes }_{\min }{C}_r^{\ast }({\mathbb{F}}_2)\to 0$$

$$0\to {\mathrm{C}}^{\ast }(G){\otimes }_{\min }K\to {\mathrm{C}}^{\ast }(G){\otimes }_{\min }{\mathrm{C}}^{\ast }(G)\to {\mathrm{C}}^{\ast }(G){\otimes }_{\min }{C}_r^{\ast }(G)\to 0$$

$$⟨f_1,f_2⟩={\int }_G\psi (g)⟨f_1(g),f_2(g)⟩dg$$

$${\int }_{G/H}{\int }_{H}f(gh)dhd\dot{g}={\int }_{G}f(g)\rho (g)dg$$

$$⟨f,g⟩={\int }_{G/H}\frac{1}{\rho (g)}⟨f_1(g),f_2(g)⟩d\dot{g}.$$

$$\Vert u\Vert :=\inf \{\Vert \xi \Vert \Vert \eta \Vert :u={\pi }_{\xi ,\eta }\},$$

$$<u,f>:={\int }_{G}u(s)f(s)ds$$

$$0\to A{\otimes }_{\min }J\to A{\otimes }_{\min }B\to A{\otimes }_{\min }C\to 0$$

$$A{\otimes }_{\min }\mathcal{B}(\mathcal{H})=A{\otimes }_{\max }\mathcal{B}(\mathcal{H})$$

$$f_{\alpha }(g)={\int }_H{\delta }_H^G(h^{-1}{)}^2{e}_{\alpha }(gh)\pi (h)\xi dh$$

$$x_{\alpha }(\dot{g})=\mathrm{sgn}(f_{\alpha }(\dot{g}))\Vert f_{\alpha }(\dot{g}){\Vert }^{1/2},$$

$$\mathrm{sgn}(\eta)=\left\{\begin{array}[]{c l}\frac{\eta}{\|\eta\|},&\textnormal{if }\eta\neq 0\\ 0,&\textnormal{if }\eta=0.\end{array}\right.$$

$$f_{\alpha }(gh)={\int }_H{\delta }_H^G(h_1^{-1}h{)}^2{e}_{\alpha }(g{h}_1)\pi (h^{-1}{h}_1)\xi d{h}_1={\delta }_H^G(h{)}^2\pi (h^{-1})f_{\alpha }(g).$$

$$({\mathrm{Ind}}_H^G\pi {)}_{x_{\alpha },x_{\alpha }}(h)\to {\pi }_{\xi ,\xi }(h)$$

$$\big{|}\left\langle\mathrm{sgn}\,x_{\alpha}(s^{-1}g),\mathrm{sgn}\,x_{\alpha}(g)\right\rangle-\pi_{\xi,\xi}(s)\big{|}<\epsilon$$

$$\Vert \pi (h)\pi (s)\xi -\pi (s)\xi \Vert <\frac{ϵ}{2}$$

$$f_{\alpha }(s^{-1}g)={\int }_H{\delta }_H^G(h^{-1}{)}^2\tau (s)e_{\alpha }(gh)\pi (h)\pi (s)\xi dh.$$

$$f_{\alpha }(s^{-1}g)$$

$$\Vert \mathrm{sgn}{f}_{\alpha }(s^{-1}g)-\pi (h_1)\pi (s)\xi \Vert <\frac{ϵ}{2}$$

$$\big{|}\left\langle\mathrm{sgn}\,x_{\alpha}(s^{-1}g),\mathrm{sgn}\,x_{\alpha}(g)\right\rangle-\pi_{\xi,\xi}(s)\big{|}<\epsilon$$

$$\Vert x_{\alpha }\Vert ={\left({\int }_{G/H}\frac{1}{\rho (g)}\Vert x_{\alpha }(g){\Vert }^{2}d\dot{g}\right)}^{1/2}\to 1.$$

$$\int_{G/H}\frac{1}{\rho(g)}\big{|}\|x_{\alpha}(s^{-1}g)\|-\|x_{\alpha}(g)\|\big{|}^{2}\,d\dot{g}\to 0$$

$$\Vert \pi (h)\pi (s)\xi -\pi (s)\xi \Vert <\frac{ϵ}{3}$$

$$\Vert \pi (h_1)f_{\alpha }(s^{-1}g)-{\int }_H{\delta }_H^G(h^{-1}{)}^2\tau (s)e_{\alpha }(gh)\pi (s)\xi dh\Vert$$

$$\left|\Vert f_{\alpha }(s^{-1}g)\Vert -{\int }_H{\delta }_H^G(h^{-1}{)}^2\tau (s)e_{\alpha }(gh)dh\right|<\frac{ϵ}{3}{\int }_H{\delta }_H^G(h^{-1}{)}^2\tau (s)e_{\alpha }(gh)dh.$$

$${\int }_{G/H}\frac{1}{\rho (g)}{\int }_H{\delta }_H^G(h^{-1}{)}^2\tau (s)e_{\alpha }(gh)dhd\dot{g}=1,$$

$$\Vert x_{\alpha }{\Vert }^2={\int }_{G/H}\frac{1}{\rho (g)}\Vert f_{\alpha }(s^{-1}g)\Vert dg\to 1.$$

$${\int }_{G/H}\frac{1}{\rho (g)}\left|{\int }_H{\delta }_H^G(h^{-1}{)}^2\tau (s)e_{\alpha }(gh)dh-{\int }_H{\delta }_H^G(h^{-1}{)}^2{e}_{\alpha }(gh)dh\right|d\dot{g}$$

$$\int_{G/H}\frac{1}{\rho(g)}\big{|}\|f_{\alpha}(s^{-1}g)\|-\|f_{\alpha}(g)\|\big{|}\,d\dot{g}\leq\epsilon.$$

$$\int_{G/H}\frac{1}{\rho(g)}\big{|}\|x_{\alpha}(s^{-1}g)\|-\|x_{\alpha}(g)\|\big{|}^{2}\,d\dot{g}\leq\int_{G/H}\frac{1}{\rho(g)}\big{|}\|x_{\alpha}(s^{-1}g)\|^{2}-\|x_{\alpha}(g)\|^{2}\big{|}\,d\dot{g}\to 0$$

Full text

1. Introduction

Let $G$ be a locally compact group, and $\pi\!:G\to\mathcal{B}(\mathcal{H}_{\pi})$ and $\sigma\!:G\to\mathcal{B}(\mathcal{H}_{\sigma})$ be (continuous unitary) representations of $G$. There are three related notions for what it means for $\pi$ to contain the representation $\sigma$. The most basic of these is when $\sigma$ is unitarily equivalent to a subrepresentation of $\pi$, but this notion is too strong for many purposes from the perspective of operator algebras. The other two notions of containment are quasi-containment and weak containment. These are the appropriate versions of containment to consider from the perspectives of von Neumann algebras and C*-algebras, respectively. Indeed, if we define $\mathrm{VN}_{\pi}:=\pi(G)^{\prime\prime}\subset\mathcal{B}(\mathcal{H}_{\pi})$ and $\mathrm{C}^{*}_{\pi}:=\overline{\pi(\mathrm{L}^{1}(G))}^{\|\cdot\|}\subset\mathcal{B}(\mathcal{H}_{\pi})$, then the identity map on $G$ extends to a (normal) $*$-homomorphism $\mathrm{VN}_{\pi}\to\mathrm{VN}_{\sigma}$ if and only if $\pi$ quasi-contains $\sigma$, and the identity map on $\mathrm{L}^{1}(G)$ extends to a $*$-homomorphism $\mathrm{C}^{*}_{\pi}\to\mathrm{C}^{*}_{\sigma}$ if and only if $\pi$ weakly contains $\sigma$.

Suppose that $H$ is a closed subgroup of a locally compact group $G$, and $\pi\!:H\to\mathcal{B}(\mathcal{H}_{\pi})$ is a representation of $H$. The main result in this paper addresses the question of when the restriction of the induced representation $(\mathrm{Ind}_{H}^{G}\pi)|_{H}$ weakly contains $\pi$. When $G$ is a discrete group, it is a straightforward exercise to check that $\pi$ is unitarily equivalent to a subrepresentation of $(\mathrm{Ind}_{H}^{G}\pi)|_{H}$ for every subgroup $H$ of $G$ and representation $\pi$ of $H$. In 1979 Cowling and Rodway showed that the restriction of the Fourier-Stieltjes algebra of $G$ to $H$, $\mathrm{B}(G)|_{H}$, is equal to $\mathrm{B}(H)$ whenever $G$ is a SIN group (see [5, Theorem 2]). An examination of their proof reveals that Cowling and Rodway actually established the following result, from which their previously stated theorem follows immediately.

Theorem 1.1** (Cowling-Rodway [5]).**

Let $G$ be a SIN group and $H$ a closed subgroup of $G$. If $\pi\!:H\to\mathcal{B}(\mathcal{H}_{\pi})$ is a representation of $H$, then $\pi$ is quasi-contained in $(\mathrm{Ind}_{H}^{G}\pi)|_{H}$.

A QSIN group is a locally compact group whose group algebra $\mathrm{L}^{1}(G)$ admits a quasi-central bounded approximate identity. This class of groups contains all amenable groups in addition to every SIN group. It is known that Theorem 1.1 fails to hold when the class of SIN groups is replaced with QSIN groups; however, the main result of this paper is that the analogue of Theorem 1.1 where quasi-containment is replaced with weak containments is true for the class of QSIN groups.

Theorem 1.2**.**

Let $G$ be a QSIN group and $H$ a closed subgroup of $G$. If $\pi\!:H\to\mathcal{B}(\mathcal{H}_{\pi})$ is a representation of $H$, then $\pi$ is weakly contained in $(\mathrm{Ind}_{H}^{G}\pi)|_{H}$.

This result provides a powerful tool for understanding local and approximation properties group C*-algebras of non-discrete groups.

Recall that Lance showed in 1973 that a discrete group $G$ is amenable if and only if $\mathrm{C}_{r}^{*}(G)$ is nuclear (see [15]). Since the quotient of a nuclear C*-algebra is nuclear, we also get the characterization that a discrete group $G$ is amenable if and only if $\mathrm{C}^{*}(G)$ is nuclear. Partially motivated by these characterizations, there is a lot of research centred around understanding the local and approximation properties of groups C*-algebras for nonamenable discrete groups. An early result in the subject due to Wasserman is that the sequence

[TABLE]

is not exact, where $J$ is the kernel of the canonical map $\mathrm{C}^{*}(\mathbb{F}_{2})\to\mathrm{C}^{*}_{r}(\mathbb{F}_{2})$ (see [21]). It follows that $\mathrm{C}^{*}(\mathbb{F}_{2})$ is not locally reflexive and $\mathrm{C}^{*}_{r}(\mathbb{F}_{2})$ does not have the local lifting property.

Lance’s characterization of amenability does not hold for nondiscrete groups since $\mathrm{C}^{*}(G)$ is nuclear for every separable connected group $G$ ([4, Corollary 6.9]). As such, the local and approximation properties of nondiscrete nonamenable groups are considered difficult to understand in general. Theorem 1.2 provides a powerful tool for understanding some of these properties for group C*-algebras of QSIN groups. As a particular application of Theorem 1.2, it is shown that the analogue of Wasserman’s result holds for every QSIN group containing a copy of $\mathbb{F}_{2}$ as a closed subgroup.

Theorem 1.3**.**

Let $G$ be a QSIN group which contains $\mathbb{F}_{2}$ as a closed subgroup. Then the sequence

[TABLE]

is not exact, where $K$ is the kernel of the canonical map $\mathrm{C}^{*}(G)\to\mathrm{C}_{r}^{*}(G)$.

Consequently $\mathrm{C}^{*}(G)$ is not locally reflexive and $\mathrm{C}^{*}_{r}(G)$ does not admit the LLP for such groups $G$.

In the final section of the paper, Theorem 1.1 and Theorem 1.2 are applied to shed light on the problem of when it is possible to “extend” a Fourier space $\mathrm{A}_{\pi}$ or “weakly extend” a Fourier-Stieltjes space $\mathrm{B}_{\pi}$ from a closed subgroup $H$ of a locally compact group $G$ to a Fourier space or Fourier-Stieltjes space of $G$.

2. Background and Notation

Due to the broad scope of topics touched on by this paper, we begin with a relatively extensive background section.

2.1. Quasi-containment and weak-containment of representations

Let $G$ be a locally compact group. A representation $\pi\!:G\to\mathcal{B}(\mathcal{H}_{\pi})$ is quasi-contained in second representation $\sigma\!:G\to\mathcal{B}(\mathcal{H}_{\sigma})$ of $G$ if $\pi$ is unitarily equivalent to a subrepresentation of some amplification of $\sigma$. Quasi-equivalence is extremely important in the study of von Neumann algebras associated to $G$ since the map $\sigma(g)\mapsto\pi(g)$ for $g\in G$ extends to a normal $*$-homomorphism from $\mathrm{VN}_{\sigma}:=\sigma(G)^{\prime\prime}\subset\mathcal{B}(\mathcal{H}_{\sigma})$ to $\mathrm{VN}_{\pi}:=\pi(G)^{\prime\prime}\subset\mathcal{B}(\mathcal{H}_{\pi})$ if and only if $\pi$ is quasi-contained in $\sigma$.

Weak containment is essential to the study of group C*-algebras in a similar way as quasi-containment is to group von Neumann algebras. For a representation $\pi\!:G\to\mathcal{B}(\mathcal{H}_{\pi})$ of $G$ and $\xi,\eta\in\mathcal{H}_{\pi}$, we will let $\pi_{\xi,\eta}\!:G\to\mathbb{C}$ be the matrix coefficient defined by $\pi_{\xi,\eta}(g)=\left\langle\rho(g)\xi,\eta\right\rangle$. A representation $\pi\!:G\to\mathcal{B}(\mathcal{H}_{\pi})$ is weakly contained in $\sigma\!:G\to\mathcal{B}(\mathcal{H}_{\sigma})$ if every positive definite function of the form $\pi_{\xi,\xi}$ for $\xi\in\mathcal{H}_{\pi}$ is the limit of positive definite functions with the form $\sum_{j=1}^{N}\sigma_{\eta_{j},\eta_{j}}$ for $N\in{\mathbb{N}}$ and $\eta_{1},\ldots,\eta_{N}\in\mathcal{H}_{\sigma}$ in the topology of uniform convergence on compact subsets of $G$. The representation $\pi$ is weakly contained in $\sigma$ if and only if $\|\pi(f)\|\leq\|\sigma(f)\|$ for every $f\in\mathrm{L}^{1}(G)$ if and only if the map $\pi(f)\mapsto\sigma(f)$ for $f\in\mathrm{L}^{1}(G)$ extends to a $*$-homomorphism from $\mathrm{C}^{*}_{\sigma}:=\overline{\sigma(\mathrm{L}^{1}(G))}^{\|\cdot\|}$ to $\mathrm{C}^{*}_{\pi}:=\overline{\pi(\mathrm{L}^{1}(G))}^{\|\cdot\|}$.

Further details on quasi-containment and weak containment can be found in [6].

2.2. Some classes of locally compact groups

A SIN group (standing for small invariant neighbourhood group) is a locally compact group $G$ whose identity admits a neighbourhood base of compact sets $K$ which are invariant under conjugation, i.e., such that $s^{-1}Ks=K$ for all $s\in G$. Examples of SIN groups include all discrete, abelian, and compact groups.

Recall that if $G$ is a locally compact group with modular function $\Delta_{G}$ and $s\in G$, then $\tau(s)\!:\mathrm{L}^{1}(G)\to\mathrm{L}^{1}(G)$ defined by $\tau(s)f(t)=f(s^{-1}ts)\Delta_{G}(s)$ is an isometric isomorphism. It is a well known result of Mosak that SIN groups are exactly the class of locally compact groups $G$ for which $\mathrm{L}^{1}(G)$ admits a central bounded approximate identity, i.e. a bounded approximate identity $\{e_{\alpha}\}\subset\mathrm{L}^{1}(G)$ such that $\tau(s)e_{\alpha}=e_{\alpha}$ for every $s\in G$ and index $\alpha$ (see [18]). Using this characterization, QSIN groups (standing for quasi-SIN groups) offer a natural generalization of SIN groups.

A locally compact group $G$ is QSIN if $\mathrm{L}^{1}(G)$ admits a quasi-central bounded approximate identity, i.e., a bounded approximate identity $\{e_{\alpha}\}\subset\mathrm{L}^{1}(G)$ such that $\|\tau(s)e_{\alpha}-e_{\alpha}\|_{1}\to 0$ uniformly on compact subsets of $G$. QSIN groups are much more general than the class of SIN groups and, by a result of Losert and Rindler, contains every amenable group (see [17]). Interested readers are encouraged to see [20] where much of the basic theory of QSIN groups is developed and [16, Remark 2.2] for a list of examples and non-examples of QSIN groups.

2.3. Induced Representations

We refer the reader to [12] as a resource on induced representations. We will be following the conventions used within this book, but pause to fix notation and summarize key results.

Let $G$ be a locally compact group and $H$ be a closed subgroup of $G$. If $\pi\!:H\to\mathcal{B}(\mathcal{H}_{\pi})$ is a representation of $H$, we set $\mathcal{F}(G,\pi)$ to be the set of norm continuous functions $f\!:G\to\mathcal{H}_{\pi}$ so that

1. (1)

$f(gh)=\delta_{H}^{G}(h)\pi(h^{-1})f(g)$ for $g\in G$ and $h\in H$, and 2. (2)

$q(\mathrm{supp}f)$ is compact in $G/H$

where $\delta_{H}^{G}(h):=\Delta_{H}(h)^{\frac{1}{2}}\Delta_{G}(h)^{-\frac{1}{2}}$ and $q\!:G\to G/H$ is the quotient map. We define an inner product on $\mathcal{F}(G,\pi)$ by letting

[TABLE]

for $f_{1},f_{2}\in\mathcal{F}(G,\pi)$, where $\psi$ is any function in $\mathrm{C}_{\mathrm{c}}(G)$ such that $\psi^{\#}(\dot{g})=1$ for every $g\in(\mathrm{supp}\,f)\cup(\mathrm{supp}\,g)$, where $\psi^{\#}\in{\mathrm{C}}_{\mathrm{c}}(G/H)$ is defined by $\psi^{\#}(\dot{g})=\int_{H}\psi(gh)\,dh$. Let $\overline{\mathcal{F}}(G,\pi)$ denote the completion of $\mathcal{F}(G,\pi)$ with respect to this inner product. The induced representation of $\pi$ is given by $\mathrm{Ind}_{H}^{G}\pi(s)f=L_{s}f$ for $f\in\mathcal{F}$, where $L_{s}$ is the continuous extension of the left shift operator $L_{s}f(g)=f(s^{-1}g)$ from $\mathcal{F}(G,\pi)$ to $\overline{\mathcal{F}}(G,\pi)$.

Fix a continuous rho-function $\rho$ for $(G,H)$, i.e., a continuous function $\rho\!:G\to(0,\infty)$ such that $\rho(gh)=\delta_{H}^{G}(h)^{2}\rho(g)$ for all $h\in H$, $g\in G$, and a quasi-invariant measure $d\dot{g}$ on $G/H$ such that

[TABLE]

for all $f\in\mathrm{C}_{\mathrm{c}}(G)$. Then a straightforward calculation shows that

[TABLE]

We will favour this latter equation in our calculations.

2.4. Fourier and Fourier-Stieltjes spaces

The basic theory of Fourier and Fourier-Stieltjes was developed by Eymard in [9] and further refined by his student Arsac in [1]. We will give an overview of the results from these two works as they apply to this paper.

Let $G$ be a locally compact group. Recall from Section 2.1 that if $\pi\!:G\to\mathcal{B}(\mathcal{H}_{\pi})$ is a representation of $G$ and $\xi,\eta\in\mathcal{H}_{\pi}$, then $\pi_{\xi,\eta}\!:G\to\mathbb{C}$ is defined by $\pi_{\xi,\eta}(s)=\left\langle\pi(s)\xi,\eta\right\rangle$. The Fourier-Stieltjes algebra $\mathrm{B}(G)$ is the set of all matrix coefficients $\pi_{\xi,\eta}$ as $\pi$ ranges over all representations of $G$ and $\xi,\eta$ over the associated Hilbert spaces $\mathcal{H}_{\pi}$. The Fourier-Stieltjes algebra is a Banach algebra which is the dual of $\mathrm{C}^{*}(G)$ with respect to pointwise operations, norm given by

[TABLE]

and dual pairing

[TABLE]

for $u\in B(G)$ and $f\in\mathrm{L}^{1}(G)$.

For each representation $\pi\!:G\to\mathcal{B}(\mathcal{H}_{\pi})$ of $G$, the Fourier space $\mathrm{A}_{\pi}$ is defined to be the norm closed linear span of matrix coefficients $\pi_{\xi,\eta}$ in $\mathrm{B}(G)$ as $\xi$ and $\eta$ range over $\mathcal{H}_{\pi}$, and the Fourier-Stieltjes space $\mathrm{B}_{\pi}$ is defined to be the weak* closure of $\mathrm{A}_{\pi}$ in $\mathrm{B}(G)$. Then $\mathrm{A}_{\pi}$ can be naturally identified with the predual of $\mathrm{VN}_{\pi}$ and $\mathrm{B}_{\pi}$ with the dual of $\mathrm{C}^{*}_{\pi}$. Every Fourier space $\mathrm{A}_{\pi}$ is invariant under both left and right translation by $G$ and, conversely, every norm closed subspace $E$ of $\mathrm{B}(G)$ which is closed under left and right translation by $G$ is a Fourier space $\mathrm{A}_{\pi}$. As a distinguished Fourier space, the Fourier algebra $\mathrm{A}(G)$ is defined to be $\mathrm{A}_{\lambda}$, where $\lambda\!:G\to\mathcal{B}(\mathrm{L}^{2}(G))$ is the left regular representation of $G$.

Let $\pi$ and $\sigma$ be representations of a locally compact group $G$. Then $\mathrm{A}_{\sigma}\subset\mathrm{A}_{\pi}$ if and only if $\sigma$ is quasi-contained in $\pi$ and $\mathrm{B}_{\sigma}\subset\mathrm{B}_{\pi}$ if and only if $\sigma$ is weakly contained in $\pi$. The Fourier space $\mathrm{A}_{\pi}$ is a subalgebra of $\mathrm{B}(G)$ if and only if $\pi\otimes\pi$ is quasi-contained in $\pi$ and, similarly, $\mathrm{B}_{\pi}$ is a subalgebra of $\mathrm{B}(G)$ if and only if $\pi\otimes\pi$ is weakly contained in $\pi$. Further, $\mathrm{A}_{\pi}$ is an ideal of $\mathrm{B}(G)$ if and only if $\pi\otimes\sigma$ is quasi-contained in $\pi$ for every representation $\sigma$ of $G$, and $\mathrm{B}_{\pi}$ is an ideal of $\mathrm{B}(G)$ if and only if $\pi\otimes\sigma$ is weakly contained in $\pi$ for every representation $\sigma$ of $G$. In particular, $\mathrm{A}(G)$ is an ideal of $\mathrm{B}(G)$ by Fell’s absorption principle.

The final facts which we will need to know about Fourier and Fourier-Stieltjes space are the following. Suppose that $H$ is a closed subgroup of a locally compact group $G$ and $\pi$ is a representation of $G$. Then $\mathrm{A}_{\pi}|_{H}=\mathrm{A}_{\pi|_{H}}$ and $\mathrm{B}_{\pi}|_{H}\subset\mathrm{B}_{\pi|_{H}}$.

2.5. Some local properties of C*-algebras

A C*-algebra (or an operator space) $A$ is locally reflexive if for every finite dimensional subspace $E\subset X$ and complete contraction $\phi\!:E\to A^{**}$, there exists a net $\phi_{i}\!:E\to A$ of complete contractions so that $\phi_{i}\to\phi$ in the point-weak* topology. Local reflexivity is also equivalent to property C*′′* (see [19, Proposition 18.15]). Recall that a C*-algebra $A$ is exact if the sequence

[TABLE]

is exact for every short exact sequence $0\to J\to B\to C\to 0$ of C*-algebras. Every exact C*-algebra is locally reflexive.

Now suppose that $A$ is a unital C*-algebra. The C*-algebra $A$ has the local lifting property (LLP) if for every finite dimensional operator system $E\subset A$, unital C*-algebra $B$ with two-sided closed ideal $J$, and unital completely positive (ucp) map $\phi\!:A\to(B/J)$ there exists a ucp map $\psi\!:E\to B$ such that $\phi|_{E}=\pi\circ\psi$ where $\pi\!:B\to B/J$ is the quotient map. We say that a non-unital C*-algebra $A$ has the LLP whenever its unitization does. Kirchberg showed that a C*-algebra $A$ has the LLP if and only if

[TABLE]

canoncially, where $\mathcal{H}$ is the separable infinite dimensional Hilbert space (see [14]).

All of this and further information on local reflexivity and the local lifting property can be found in each of the books [3, 8, 19].

3. Main result

Theorem 3.1**.**

Suppose $G$ is a QSIN group and $H$ is a closed subgroup of $G$. Then $(\mathrm{Ind}_{H}^{G}\pi)|_{H}$ weakly contains $\pi$ for every representation $\pi\!:H\to\mathcal{B}(\mathcal{H}_{\pi})$ of $H$.

Proof.

Fix a unit vector $\xi\in\mathcal{H}_{\pi}$ and recall from Section 2.2 that $\tau(s)\!:\mathrm{L}^{1}(G)\to\mathrm{L}^{1}(G)$ is defined by $\tau(s)f(t)=f(s^{-1}ts)\Delta_{G}(s)$ for every $s\in G$ and $f\in\mathrm{L}^{1}(G)$. Since $G$ is QSIN, there exists a net $\{e_{\alpha}\}$ in $\mathrm{L}^{1}(G)_{1}^{+}\cap\mathrm{C}_{\mathrm{c}}(G)$ such that $\mathrm{supp}\,e_{\alpha}\to\{e\}$ and $\|\tau(s)e_{\alpha}-e_{\alpha}\|_{1}\to 0$ uniformly on compact subsets of $G$ (see [20, Theorem 2.6]). For each index $\alpha$, we define $f_{\alpha}\in\mathrm{C}(G,\mathcal{H}_{\pi})$ by

[TABLE]

and $x_{\alpha}\in\mathrm{C}(G,\mathcal{H}_{\pi})$ by

[TABLE]

where $\mathrm{sgn}\!:\mathcal{H}_{\pi}\to\mathcal{H}_{\pi}$ is given by

[TABLE]

Observe that $x_{\alpha}\in\mathcal{F}(G,\mathcal{H}_{\pi})$ since

[TABLE]

We will show that

[TABLE]

uniformly on compact subsets of $H$.

Claim 1*.*

Let $K$ be a compact subset of $H$ and $\epsilon>0$. There exists an index $\alpha_{0}$ so that if $s\in K$, $g\in G$, and $\alpha\geq\alpha_{0}$, then

[TABLE]

whenever $x_{\alpha}(g)\neq 0$ and $x_{\alpha}(s^{-1}g)\neq 0$.

Proof of Claim. Without loss of generality, we may assume that $\epsilon<2\sqrt{2}$ and $e\in K$. Since $\pi(K)\xi$ is a compact subset of $\mathcal{H}_{\pi}$, there exists a symmetric neighbourhood $U$ of the identity in $G$ so that

[TABLE]

for all $h\in U^{2}\cap H$ and $s\in K$. Further, since $K$ is compact and $\mathrm{supp}\,e_{\alpha}\to\{e\}$, there exists an index $\alpha_{0}$ so that \mathrm{supp}\big{(}\tau(s)\,e_{\alpha}\big{)}\subset U for all $s\in K$ and $\alpha\geq\alpha_{0}$. We will show that this index $\alpha_{0}$ produces the desired result.

Let $\alpha\geq\alpha_{0}$. A routine calculation shows that

[TABLE]

If $\tau(s)e_{\alpha}(gh)\neq 0$ for some $h\in H$, then $g=g_{1}h_{1}$ for some $g_{1}\in U$ and $h_{1}\in H$. Hence,

[TABLE]

Since $\|\pi(h)\pi(s)\xi-\pi(s)\xi\|<\frac{\epsilon}{2}<\sqrt{2}$ for every $h\in U^{2}\cap H$,

[TABLE]

for all $s\in K$ which satisfy $f_{\alpha}(s^{-1}g)\neq 0$. Thus,

[TABLE]

for all $s\in K$ and $g\in G$ which satisfy $x_{\alpha}(g)\neq 0$ and $x_{\alpha}(s^{-1}g)\neq 0$. $\blacksquare$

Claim 2*.*

Let $K$ be a compact subset of $H$. Then

• (a)

[TABLE]

• (b)

[TABLE]

uniformly for $s\in K$.

Proof of Claim. Without loss of generality, we may assume that $e\in K$. Choose a symmetric neighbourhood $U$ of the identity in $G$ so that

[TABLE]

for all $s\in K$ and $h\in U^{2}\cap H$, and an index $\alpha_{0}$ so that $\mathrm{supp}\,\tau(s)e_{\alpha}\subset U$ for all $s\in K$ and $\alpha\geq\alpha_{0}$.

Fix $\alpha\geq\alpha_{0}$ and $g\in G$. If $g=g_{1}h_{1}$ for some $g_{1}\in U$ and $h_{1}\in H$, then

[TABLE]

In particular, this implies that

[TABLE]

Since

[TABLE]

it follows immediately that

[TABLE]

This establishes part (a) of the claim.

Next observe that

[TABLE]

Since $\|\tau(s)e_{\alpha}-e_{\alpha}\|_{1}\to 0$ uniformly for $s\in K$, by replacing $\alpha_{0}$ with a higher index if necessary, we may assume that $\|\tau(s)e_{\alpha}-e_{\alpha}\|_{1}<\frac{\epsilon}{3}$ for all $s\in K$ and $\alpha\geq\alpha_{0}$. Combining this with Equation 1 gives

[TABLE]

Hence,

[TABLE]

uniformly for $s\in K$. This establishes part (b) of the claim. $\blacksquare$

Claim 3*.*

Let $K$ be a compact subset of $H$. Then $(\mathrm{Ind}_{H}^{G}\pi)_{x_{\alpha},x_{\alpha}}(h)\to\pi_{\xi,\xi}(s)$ uniformly for $s\in K$.

Proof of Claim. We begin by observing that if $s\in H$, then

[TABLE]

By applying Claims 1 and 2 to this equation, we conclude that $(\mathrm{Ind}_{H}^{G}\pi)_{x_{\alpha},x_{\alpha}}(h)\to\pi_{\xi,\xi}(s)$ uniformly for $s\in K$. $\blacksquare$

∎

4. Local properties of group C*-algebras

In 1976 Wasserman gave the first example of a C*-algebra which is not exact.

Theorem 4.1** (Wasserman [21]).**

The sequence

[TABLE]

is not exact, where $J$ is the kernel of the canonical map $\mathrm{C}^{*}(\mathbb{F}_{2})\to\mathrm{C}^{*}_{r}(\mathbb{F}_{2})$.

In 1985 Effros and Haagerup proved the following two results in [7].

Theorem 4.2** (Effros-Haagerup [7, Proposition 5.3]).**

If $A$ is a locally reflexive C-algebra, then the sequence*

[TABLE]

is exact for every closed two-sided ideal $J$ of $A$ and every C-algebra $C$.*

Theorem 4.3** (Effros-Haagerup [7, Theorem 3.2]).**

Let $B$ be a C-algebra and $J$ a closed two sided ideal of $B$. If $A:=B/J$ has the local lifting property, then the sequence*

[TABLE]

is exact for every C-algebra $C$.*

Consequently, $\mathrm{C}^{*}(\mathbb{F}_{2})$ is not locally reflexive and $\mathrm{C}^{*}_{r}(\mathbb{F}_{2})$ does not admit the LLP.

We will now show that the analogue of Theorem 4.1 holds when $\mathbb{F}_{2}$ is replaced with a QSIN group $G$ containing a closed copy of $\mathbb{F}_{2}$. As an immediate consequence, we will have that $\mathrm{C}^{*}(G)$ is not locally reflexive and $\mathrm{C}^{*}_{r}(G)$ does not admit the LLP. Before proceeding to the proof, we note that a very simple argument achieves this result when $G$ is discrete.

Remark 4.4*.*

Let $G$ be a discrete group containing a copy of $\mathbb{F}_{2}$. Then $\mathrm{C}^{*}(G)$ is not locally reflexive since local reflexivity passes to subspaces.

Next let $P$ be the orthogonal projection from $\ell^{2}(G)$ onto $\ell^{2}(\mathbb{F}_{2})$ and observe that $\Phi\!:\mathrm{C}^{*}_{r}(G)\to\mathrm{C}^{*}_{r}(\mathbb{F}_{2})$ defined by $\Phi(a)=PaP$ is a conditional expectation. Thus if $\phi\!:\mathrm{C}^{*}_{r}(\mathbb{F}_{2})\to B$ is any ucp map into a C*-algebra $B$, then $\tilde{\phi}:=\phi\circ\Phi$ is a ucp map from $\mathrm{C}^{*}_{r}(G)$ to $B$ which extends $\phi$. This implies that if $\mathrm{C}^{*}_{r}(G)$ had the LLP, then so would $\mathrm{C}^{*}_{r}(\mathbb{F}_{2})$. Since $\mathrm{C}^{*}_{r}(\mathbb{F}_{2})$ does not have the LLP, we conclude that neither does $\mathrm{C}^{*}_{r}(G)$.

Lemma 4.5**.**

Let $G_{1}$ and $G_{2}$ be groups, and $H_{1}$ a closed subgroup of $G_{1}$. If $\pi$ is a representation of $H_{1}\times G_{2}$, then

1. (a)

$(\mathrm{Ind}_{H_{1}\times G_{2}}^{G_{1}\times G_{2}}\pi)|_{G_{1}}$* is unitarily equivalent to $\mathrm{Ind}_{H_{1}}^{G_{1}}(\pi|_{H_{1}})$,* 2. (b)

$(\mathrm{Ind}_{H_{1}\times G_{2}}^{G_{1}\times G_{2}}\pi)|_{G_{2}}$* is quasi-contained in $\pi|_{G_{2}}$.*

Proof.

(a) Fix a quasi-invariant measure $d\dot{g}_{1}$ on $G_{1}/H_{1}$. This becomes a quasi-invariant measure on $(G_{1}\times G_{2})/(H_{1}\times G_{2})$ when we identify $(G_{1}\times G_{2})/(H_{1}\times G_{2})$ with $G_{1}/H_{1}$. It is straightforward to verify that $U\!:\mathcal{F}(G_{1},\pi|_{H_{1}})\to\mathcal{F}(G_{1}\times G_{2},\pi)$ defined by

[TABLE]

is a surjective intertwining isometry.

(b) Let $f\in\mathrm{C}_{\mathrm{c}}(G_{1})$ and $\xi\in\mathcal{H}_{\pi}$. Then $x\!:G_{1}\times G_{2}\to\mathcal{H}_{\pi}$ defined by

[TABLE]

is in $\mathcal{F}(G,\mathcal{H}_{\pi})$ by the identification of $\mathcal{F}(G_{1},\pi|_{H_{1}})$ with $\mathcal{F}(G_{1}\times G_{2},\pi)$ established above. A straightforward calculation shows that

[TABLE]

where

[TABLE]

In particular $(\mathrm{Ind}_{H_{1}\times G_{2}}^{G_{1},G_{2}}\pi)_{x,x}|_{G_{2}}\in\mathrm{A}_{\pi|_{G_{2}}}$. Since the linear span of functions with the form of $x$ (as $f$ ranges over $\mathrm{C}_{\mathrm{c}}(G_{1})$ and $\xi$ over $\mathcal{H}_{\pi}$) is a dense subspace of $\overline{\mathcal{F}}(G_{1}\times G_{2},\pi)$ (see [12, Lemma 2.24]), it follows that $\mathrm{A}_{(\mathrm{Ind}\,\pi)|_{G_{2}}}\subset\mathrm{A}_{\pi|_{G_{2}}}$. Thus $(\mathrm{Ind}_{H_{1}\times G_{2}}^{G_{1}\times G_{2}}\pi)|_{G_{2}}$ is quasi-contained in $\pi|_{G_{2}}$. ∎

Recall that $\mathrm{L}^{1}(G\times G)=\mathrm{L}^{1}(G)\otimes_{\gamma}\mathrm{L}^{1}(G)$, where $\otimes_{\gamma}$ denotes the Banach space projective tensor product. As such, we may view C*-completions of tensor products of group C*-algebras of $G$ as being C*-completions of $\mathrm{L}^{1}(G\times G)$. This is used implicitly in the following theorem when we identify $*$-representations of tensor products of group C*-algebras of $G$ with representations of $G\times G$.

Theorem 4.6**.**

Let $G$ be a QSIN group which contains $\mathbb{F}_{2}$ as a closed subgroup. Then the sequence

[TABLE]

is not exact, where $K$ is the kernel of the canonical map $\mathrm{C}^{*}(G)\to\mathrm{C}_{r}^{*}(G)$.

Proof.

Wasserman’s result above is equivalent to the statement that the quotient map

[TABLE]

is not injective. Let \sigma\!:\mathbb{F}_{2}\times\mathbb{F}_{2}\to\big{(}\mathrm{C}^{*}(\mathbb{F}_{2})\otimes_{\min}\mathrm{C}^{*}(\mathbb{F}_{2})\big{)}/\big{(}\mathrm{C}^{*}(\mathbb{F}_{2})\otimes_{\min}J\big{)} be the canonical map, $\pi_{u}\!:\mathbb{F}_{2}\to\mathcal{B}(\mathcal{H}_{u})$ be the universal representation of $\mathbb{F}_{2}$, and $\lambda_{\mathbb{F}_{2}}\!:\mathbb{F}_{2}\to\mathcal{B}(\ell^{2}(\mathbb{F}_{2}))$ the left regular representation of $\mathbb{F}_{2}$. Then $\pi_{u}\times\lambda_{\mathbb{F}_{2}}$ does not weakly contain $\sigma$. This will be used in our proof that sequence 2 is not exact.

Consider the representation $\omega:=\mathrm{Ind}_{\mathbb{F}_{2}\times\mathbb{F}_{2}}^{G\times G}\sigma$. Observe that $\omega$ is weakly contained in $\mathrm{Ind}_{\mathbb{F}_{2}}^{G}\pi_{u}\times\mathrm{Ind}_{\mathbb{F}_{2}}^{G}\pi_{u}$ since $\sigma$ is weakly contained in $\mathrm{Ind}_{\mathbb{F}_{2}\times\mathbb{F}_{2}}^{G\times G}(\pi_{u}\times\pi_{u})$ and $\mathrm{Ind}_{\mathbb{F}_{2}\times\mathbb{F}_{2}}^{G\times G}(\pi_{u}\times\pi_{u})=\mathrm{Ind}_{\mathbb{F}_{2}}^{G}\pi_{u}\times\mathrm{Ind}_{\mathbb{F}_{2}}^{G}\pi_{u}$ (see [12, Theorem 2.53]). So $\omega$ extends to a $*$-representation of $\mathrm{C}^{*}(G)\otimes_{\min}\mathrm{C}^{*}(G)$. Notice that $\omega|_{\{e\}\times G}$ is weakly contained in $\lambda_{G}$ since

[TABLE]

is quasi-contained in $\mathrm{Ind}_{\mathbb{F}_{2}}^{G}(\sigma|_{\{e\}\times G})$ by Lemma 4.5 and $\mathrm{Ind}_{\mathbb{F}_{2}}^{G}(\sigma|_{\{e\}\times G})$ is weakly equivalent to $\mathrm{Ind}_{\mathbb{F}_{2}}^{G}\lambda_{\mathbb{F}_{2}}=\lambda_{G}$, where $\lambda_{G}$ is the left regular representation of $G$. So the kernel of the representation $\omega$ when applied to $\mathrm{C}^{*}(G)\otimes_{\min}\mathrm{C}^{*}(G)$ contains $\mathrm{C}^{*}(G)\otimes_{\min}K$. If sequence 2 were exact, then $\omega$ would be weakly contained in $\pi_{u,G}\times\lambda_{G}$, where $\pi_{u,G}$ is the universal representation of $G$. Then $\omega|_{\mathbb{F}_{2}\times\mathbb{F}_{2}}$ would be weakly contained in $(\pi_{u,G}\times\lambda_{G})|_{\mathbb{F}_{2}\times\mathbb{F}_{2}}$, which is weakly contained in $\pi_{u}\times\lambda_{\mathbb{F}_{2}}$ since $\pi_{u,G}|_{\mathbb{F}_{2}}$ is quasi-contained in $\pi_{u}$ and $\lambda_{G}|_{\mathbb{F}_{2}}$ is quasi-equivalent to $\lambda_{\mathbb{F}_{2}}$ by the Herz restriction theorem (see [11]). This, however, is a contradiction since $\omega|_{\mathbb{F}_{2}\times\mathbb{F}_{2}}$ weakly contains $\sigma$ by Theorem 3.1 and $\sigma$ is not weakly contained in $\omega|_{\mathbb{F}_{2}}$. Thus sequence 2 is not exact. ∎

Corollary 4.7**.**

If $G$ is a QSIN group which contains $\mathbb{F}_{2}$ as a closed subgroup, then $\mathrm{C}^{*}(G)$ is not locally reflexive and $\mathrm{C}^{*}_{r}(G)$ does not have the LLP.

5. (Weak) Extensions of Fourier(-Stieltjes) spaces

Let $G$ be a locally compact group and $H$ a closed subgroup of $G$. We say that Fourier space $\mathrm{A}_{\pi}$ of $H$ extends to a Fourier space of $G$ if there exists a Fourier space $\mathrm{A}_{\sigma}$ of $G$ such that $\mathrm{A}_{\sigma}|_{H}=\mathrm{A}_{\pi}$. Equivalently, $\mathrm{A}_{\pi}$ extends to a Fourier space of $G$ if and only if there exists a representation $\sigma$ of $G$ such that $\sigma|_{H}$ is quasi-equivalent to $G$ if and only if $\mathrm{VN}_{\pi}=\mathrm{VN}_{\sigma|_{H}}$ canonically. Similarly we say that a Fourier-Stieltjes space $\mathrm{B}_{\pi}$ of $H$ weakly extends to a Fourier-Stieltjes space of $G$ if there exists a Fourier-Stiejtes space $\mathrm{B}_{\sigma}$ of $G$ such that $\mathrm{B}_{\sigma}|_{H}$ is a weak* dense subspace of $\mathrm{B}_{\pi}$. Equivalently, $\mathrm{B}_{\pi}$ weakly extends to a Fourier-Stieltjes space of $G$ if and only if there exists a representation $\sigma$ of $G$ such that $\sigma|_{H}$ is weakly equivalent to $\pi$ if and only if $\mathrm{C}^{*}_{\pi}=\mathrm{C}^{*}_{\sigma|_{H}}$ canonically.

In this section we will look at when a Fourier(-Stieltjes) space $\mathrm{A}_{\pi}$ (resp. $\mathrm{B}_{\pi}$) of a closed subgroup $H$ of a locally compact group $G$ (weakly) extends to a Fourier(-Stieltjes) space of $G$. We will focus our attention when $\mathrm{A}_{\pi}$ (resp. $\mathrm{B}_{\pi}$) is an ideal of the Fourier-Stieltjes algebra $\mathrm{B}(H)$. The majority of well studied Fourier and Fourier-Stieltjes spaces fall into this category. The special case of the extension problem when the Fourier space $\mathrm{A}_{\pi}$ is taken to be the Fourier algebra $\mathrm{A}(H)$ or the Fourier-Stieltjes algebra $\mathrm{B}(H)$ are already well studied. The Herz restriction theorem immediately implies that the Fourier algebra $\mathrm{A}(H)$ always extends to a Fourier space of $G$ for every locally compact group $G$, and a previously mentioned result of Cowling and Rodway gives that the Fourier-Stieltjes algebra $\mathrm{B}(H)$ extends to a Fourier space of $G$ whenever $G$ is a SIN group.

Theorem 5.1** (Herz Restriction Theorem [11]).**

Let $H$ be a closed subgroup of a locally compact group $G$. Then $\mathrm{A}(G)|_{H}=\mathrm{A}(H)$.

Theorem 5.2** (Cowling-Rodway [5]).**

Let $H$ be a closed subgroup of a SIN group $G$. Then $\mathrm{B}(G)|_{H}=\mathrm{B}(H)$.

As a further example of a case which has been studied, Ghandehari has shown that the analogue of Theorem 5.2 holds for the Rajchman algebra $\mathrm{B}_{0}(G)$ (see [10]).

We observe that the weak analogue of Theorem 5.2 for the class of QSIN groups is an immediate consequence of Theorem 3.1.

Corollary 5.3**.**

Let $H$ be a closed subgroup of a QSIN group $G$. Then $\mathrm{B}(G)|_{H}$ is weak dense in $\mathrm{B}(H)$.*

Let $G={\mathbb{R}}\rtimes{\mathbb{R}}^{+}$ be the $ax+b$ group and $H$ be the subgroup ${\mathbb{R}}$. Then $G$ is a QSIN group by virtue of being amenable, but $\mathrm{B}(G)|_{H}=\mathrm{A}({\mathbb{R}})\oplus\mathbb{C}1_{\mathbb{R}}\subsetneq\mathrm{B}({\mathbb{R}})$ (see [13]). Thus the conclusion of Corollary 5.3 cannot be replaced with $\mathrm{B}(G)|_{H}=\mathrm{B}(H)$. Further, there are also counterexamples to Corollary 5.3 when we drop the assumption that $G$ is QSIN (see [2, Lemma 3]).

We now characterize when an arbitrary Fourier(-Stieltjes) space of a closed subgroup $H$ of a (Q)SIN group $G$ extends to a Fourier(-Stieltjes) space of $G$.

Lemma 5.4**.**

Let $G$ be a locally compact group, $H$ a closed subgroup of $G$, and $\pi$ a representation of $G$.

• (a)

Suppose $G$ is a SIN group and $\mathrm{A}_{\pi}$ is an ideal of $\mathrm{B}(H)$. If $\mathrm{A}_{\pi}$ extends to a Fourier space of $G$, then there exists a representation $\sigma$ of $G$ such that $\mathrm{A}_{\sigma}|_{H}=\mathrm{A}_{\pi}$ and $\mathrm{A}_{\sigma}$ is an ideal of $\mathrm{B}(G)$.

• (b)

Suppose $G$ is a QSIN group and $\mathrm{B}_{\pi}$ is an ideal of $\mathrm{B}(H)$. If $\mathrm{B}_{\pi}$ weakly extends to a Fourier-Stieltjes space of $G$, then there exists a representation of $\sigma$ of $G$ such that $\mathrm{B}_{\sigma}|_{H}$ is weak dense subspace of $\mathrm{B}_{\pi}$.*

Proof.

We will prove (b). The proof of (a) is similar but easier.

Suppose that $\sigma^{\prime}$ be a representation of $G$ such that $\mathrm{B}_{\sigma^{\prime}}|_{H}$ is a weak* dense subspace of $\mathrm{B}_{\pi}$. Define

[TABLE]

Then $E$ is a translation invariant ideal of $\mathrm{B}(G)$ and, so, the weak* closure of $E$ is equal to some Fourier-Stieltjes space $\mathrm{B}_{\sigma}$ of $G$ where $\mathrm{B}_{\sigma}$ is an ideal of $\mathrm{B}(G)$. Next observe that the weak* closure of $E|_{H}$ in $\mathrm{B}(H)$ is $\mathrm{B}_{\pi}$ since $\mathrm{A}_{\sigma^{\prime}}|_{H}\subset E|_{H}\subset\mathrm{B}_{\pi}$. Thus $\mathrm{B}_{\sigma}|_{H}$ is a weak* dense subset of $\mathrm{B}_{\pi}$. ∎

Proposition 5.5**.**

Let $G$ be a locally compact group, $H$ a closed subgroup of $G$, and $\pi$ a representation of $G$.

1. (a)

Suppose $G$ is a SIN group and $\mathrm{A}_{\pi}$ is an ideal of $\mathrm{B}(G)$. Then $\mathrm{A}_{\mathrm{Ind}_{H}^{G}\pi}$ is the smallest Fourier space of $G$ which is an ideal of $\mathrm{B}(G)$ and whose restriction to $H$ contains $\mathrm{A}_{\pi}$. 2. (b)

Suppose $G$ is a QSIN group and $\mathrm{B}_{\pi}$ is an ideal of $\mathrm{B}(G)$. Then $\mathrm{B}_{\mathrm{Ind}_{H}^{G}\pi}$ is the smallest Fourier-Stieltjes space of $G$ which is an ideal of $\mathrm{B}(G)$ and whose restriction to $H$ contains $\mathrm{B}_{\pi}$ in its weak closure in $\mathrm{B}(H)$.*

Proof.

We will show (b). The proof of (a) follows by replacing Theorem 3.1 with Theorem 1.1 and weak containment by quasi-containment.

By Theorem 3.1, $\pi$ is weakly contained in $(\mathrm{Ind}_{H}^{G}\pi)|_{H}$. So $\mathrm{B}_{\pi}\subset\mathrm{B}_{(\mathrm{Ind}_{H}^{G}\pi)|_{H}}$. Since $\mathrm{A}_{(\mathrm{Ind}_{H}^{G}\pi)|_{H}}\subset\mathrm{B}_{(\mathrm{Ind}_{H}^{G}\pi)}|_{H}$, we deduce that $\mathrm{B}_{\pi}\subset\overline{\mathrm{B}_{\mathrm{Ind}_{H}^{G}\pi}|_{H}}^{w^{*}}$. Next suppose that $\sigma$ is a representation of $G$ such that $\mathrm{B}_{\sigma}$ is an ideal of $\mathrm{B}(G)$ and $\overline{(\mathrm{B}_{\sigma}|_{H})}^{w^{*}}\supset\mathrm{B}_{\pi}$. Then $\sigma\otimes\mathrm{Ind}_{H}^{G}1_{H}\prec\sigma$ since $\mathrm{B}_{\sigma}$ is an ideal of $\mathrm{B}(G)$. Further,

[TABLE]

(see [12, Theorem 2.58]) since $\pi\prec\sigma|_{H}$. Thus $\mathrm{Ind}_{H}^{G}\pi\prec\sigma$ or, equivalently, $\mathrm{B}_{\mathrm{Ind}_{H}^{G}\pi}\subset\mathrm{B}_{\sigma}$. Finally, $\mathrm{B}_{\mathrm{Ind}_{H}^{G}\pi}$ is an ideal of $\mathrm{B}(G)$ since $\sigma\otimes\mathrm{Ind}_{H}^{G}\pi=\mathrm{Ind}_{H}^{G}(\sigma|_{H}\otimes\pi)\prec\mathrm{Ind}_{H}^{G}\pi$ for every representation $\sigma$ of $G$.

∎

Corollary 5.6**.**

Let $G$ be a locally compact group, $H$ a closed subgroup of $G$, and $\pi$ a representation of $H$.

1. (a)

Suppose $G$ is a SIN group and $\mathrm{A}_{\pi}$ is an ideal of $\mathrm{B}(H)$. The following are equivalent.

1. (i)

$\mathrm{A}_{\pi}$* extends to a Fourier space of $G$;* 2. (ii)

$\mathrm{A}_{\mathrm{Ind}_{H}^{G}\pi}|_{H}\subset\mathrm{A}_{\pi}$; 3. (iii)

$\mathrm{A}_{\mathrm{Ind}_{H}^{G}\pi}|_{H}=\mathrm{A}_{\pi}$. 2. (b)

Suppose that $G$ is a QSIN group and $\mathrm{B}_{\pi}$ is an ideal of $\mathrm{B}(H)$. The following are equivalent.

1. (i)

$\mathrm{B}_{\pi}$* weakly extends to a Fourier-Stieltjes space of $G$;* 2. (ii)

$\mathrm{B}_{\mathrm{Ind}_{H}^{G}\pi}|_{H}\subset\mathrm{B}_{\pi}$; 3. (iii)

$\mathrm{B}_{(\mathrm{Ind}_{H}^{G}\pi)|_{H}}=\mathrm{B}_{\pi}$.

We note that Corollary 5.6 need not hold when the condition that $\mathrm{A}_{\pi}$ is simply a subalgebra of $\mathrm{B}(H)$. Indeed, suppose that $H$ is a non-normal closed subgroup of a locally compact group $G$. Then $\mathrm{A}_{1_{H}}$ is subalgebra of $\mathrm{B}(G)$ which clearly extends to a Fourier space of $G$ despite the fact that $(\mathrm{Ind}_{H}^{G}1_{H})|_{H}$ is not quasi-contained in $1_{H}$.

We end this paper with an example which shows that not every Fourier(-Stieltjes) space of a closed subgroup $H$ of a (Q)SIN group $G$ (weakly) extends to Fourier(-Stieltjes) space of $G$.

Example 5.7**.**

Let $G_{1}$ be any noncompact locally compact group. Then $Z/2\mathbb{Z}$ acts on $G_{1}\times G_{1}$ by switching coordinates. Consider the group $G:=(G_{1}\times G_{1})\rtimes(\mathbb{Z}/2\mathbb{Z})$ with closed subgroup $H:=G_{1}\times G_{1}$.

Define a subspace $E$ of the Fourier-Stietjes algebra $\mathrm{B}(H)$ by

[TABLE]

Then $E$ is a norm closed translation invariant ideal of $\mathrm{B}(G)$, i.e, $E$ is a Fourier space which is an ideal of $\mathrm{B}(H)$. However $E$ does not extend to a Fourier space of $G$. Indeed, if $\mathrm{A}_{\sigma}$ is a Fourier space of $G$ such that $\mathrm{A}_{\sigma}|_{H}\supset E$, then $\mathrm{A}_{\sigma}|_{G_{1}\times\{e\}}=\mathrm{B}(G_{1}\times\{e\})\supsetneq\mathrm{A}(G_{1}\times\{e\})$ since $\mathrm{A}_{\sigma}$ is translation invariant and $u\times v\in E$ for all $u\in\mathrm{A}(G_{1})$ and $v\in\mathrm{B}(G_{1})$.

The same analysis as above shows that if $G_{1}$ is any nonamenable group, and $G$ and $H$ are defined as above, then

[TABLE]

(where $\mathrm{B}_{r}(H)$ denotes the reduced Fourier-Stieltjes algebra of $H$) is a Fourier-Stieltjes space of $H$ which is an ideal in $\mathrm{B}(H)$ but does not weakly extend to a Fourier-Stieltjes space of $G$.