Representations

Abstract

In this chapter we introduce the basic concepts of representation theory of locally compact groups. Classically, a representation of a group G is an injective group homomorphism from G to some ${\rm GL}_n({\mathbb C})$, the idea being that the “abstract” group G is “represented” as a matrix group.

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In this chapter we introduce the basic concepts of representation theory of locally compact groups. Classically, a representation of a group G is an injective group homomorphism from G to some ${\rm GL}_n({\mathbb C})$, the idea being that the “abstract” group G is “represented” as a matrix group.

In order to understand a locally compact group, it is necessary to consider its actions on possibly infinite dimensional spaces like $L^2(G)$. For this reason, one considers infinite dimensional representations as well.

6.1 Schur’s Lemma

For a Banach space V, let ${\rm GL}_{\rm cont}(V)$ be the set of bijective bounded linear operators T on V. It follows from the Open Mapping Theorem C.1.5 that the inverses of such operators are bounded as well, so that ${\rm GL}_{\rm cont}(V)$ is a group. Let G be a topological group. A representation of G on a Banach space V is a group homomorphism of G to the group ${\rm GL}_{\rm cont}(V)$, such that the resulting map $G\times V\to V$, given by $(g,v)\ \mapsto\ \pi(g)v,$ is continuous.

Lemma 6.1.1

Let π be a group homomorphism of the topological group G to ${\rm GL}_ {\rm cont}(V)$ for a Banach space V. Then π is a representation if and only if

• (a) the map $g\mapsto \pi(g)v$ is continuous at g = 1 for every $v\in V,$ and

• (b) the map $g\mapsto\Vert{\pi(g)}\Vert_{\rm op}$ is bounded in a neighborhood of the unit in G.

Proof

Suppose π is a representation. Then (a) is obvious. For (b) note that for every neighborhood Z of zero in V there exists a neighborhood Y of zero in V and a neighborhood U of the unit in G such that $\pi(U)Y\subset Z$. This proves (b). For the converse direction write

$$\begin{aligned} \Vert{\pi(g)v-\pi(g_0)v_0\Vert} &\le \Vert{\pi(g_0)}\Vert_{\rm op}\Vert{\pi(g_0^{-1}g)v- v_0}\Vert\\ &\le \Vert{\pi(g_0)}\Vert_{\rm op}\Vert{\pi(g_0^{-1}g)(v-v_0)}\Vert\\ & + \Vert{\pi(g_0)}\Vert_{\rm op}\Vert{\pi(g_0^{-1}g)v_0-v_0}\Vert\\ &\le \Vert{\pi(g_0)}\Vert_{\rm op}\Vert{\pi(g_0^{-1}g)}\Vert_{\rm op}\Vert{v-v_0}\Vert\\ & + \Vert{\pi(g_0)}\Vert_{\rm op}\Vert{\pi(g_0^{-1}g)v_0-v_0}\Vert.\end{aligned}$$

Under the assumptions given, both terms on the right are small if g is close to g ₀, and v is close to v ₀. □

Examples 6.1.2

• For a continuous group homomorphism $\chi: G\to{\mathbb C}^\times$ define a representation $\pi_\chi$ on $V={\mathbb C}$ by $\pi_\chi(g)v=\chi(g)\cdot v$.

• Let $G={\rm SL}_2({\mathbb R})$ be the group of real 2 × 2 matrices of determinant one. This group has a natural representation on ${\mathbb C}^2$ given by matrix multiplication.

Definition

Let V be a Hilbert space. A representation π on V is called a unitary representation if $\pi(g)$ is unitary for every. That means π is unitary if ${\left\langle{\pi(g)v,\pi(g)w}\right\rangle}={\left\langle{v,w}\right\rangle}$ holds for every $g\in G$ and all $v,w\in V$.

Lemma 6.1.3

A representation π of the group G on a Hilbert space V is unitary if and only if $\pi(g^{-1})=\pi(g)^*$ holds for every $g\in G$.

Proof

An operator T is unitary if and only if it is invertible and $T^*=T^{-1}$. For a representation π and $g\in G$ the operator $\pi(g)$ is invertible and satisfies $\pi(g^{-1})=\pi(g)^{-1}$. So π is unitary if and only if for every $g\in G$ one has $\pi(g^{-1})=\pi(g)^{-1}=\pi(g)^*$. □

Examples 6.1.4

• The representation $\pi_\chi$ defined by a continuous group homomorphism $\chi: G\to{\mathbb C}^\times$ is unitary if and only if χ maps into the compact torus ${\mathbb T}$.

• Let G be a locally compact group. On the Hilbert space $L^2(G)$ consider the left regular representation $x\mapsto L_x$ with

$$L_x\phi(y)= \phi(x^{-1}y),\qquad \phi\in L^2(G).$$

This representation is unitary, as by the left invariance of the Haar measure,

$$\begin{aligned} {\left\langle{L_x\phi,L_x,\psi}\right\rangle} &= \int_G L_x\phi(y){\overline{L_x\psi(y)}}\,dy\\ &= \int_G\phi(x^{-1}y){\overline{\psi(x^{-1}y)}}\,dy\\ &= \int_G \phi(y){\overline{\psi(y)}}\,dy={\left\langle{\phi,\psi}\right\rangle}.\end{aligned}$$

Definition

Let $(\pi_1,V_1)$ and $(\pi_2,V_2)$ be two unitary representations. On the direct sum $V=V_1\oplus V_2$ one has the direct sum representation $\pi=\pi_1\oplus\pi_2$. More generally, if $\{\pi_i:i\in I\}$ is a family of unitary representations acting on the Hilbert spaces V _i , we write ${\bigoplus}_{i\in I}\pi_i$ for the direct sum of the representations π _i , $i\in I$ on the Hilbert space ${\widehat{\bigoplus}}_{i\in I}V_i$. See also Exercise 6.3 and appendix C.3.

Example 6.1.5

Let $G={\mathbb R}/{\mathbb Z}$, and let $V=L^2({\mathbb R}/{\mathbb Z})$. Let π be the left regular representation. By the Plancherel Theorem, the elements of the dual group $\widehat{G}=\{e_k: k\in{\mathbb Z}\}$ with $e_k([x])=e^{2\pi\! ikx}$ form an orthonormal basis of $L^2({\mathbb R}/{\mathbb Z})$ so that π is a direct sum representation on $V={\widehat{\bigoplus}}_{k\in{\mathbb Z}}{\mathbb C} e_k$, where $e_k(x)=e^{2\pi\! ikx}$ and G acts on ${\mathbb C} e_k$ through the character $\bar e_k$.

Definition

A representation $(\pi,V_\pi)$ is called a subrepresentation of a representation $(\eta,V_\eta)$ if $V_\pi$ is a closed subspace of $V_\eta$ and π equals η restricted to $V_\pi$. So every closed subspace $U\subset V_\eta$ that is stable under η, i.e., $\eta(G)U\subset U$, gives rise to a subrepresentation.

A representation is called irreducible if it does not possess any proper subrepresentation, i.e., if for every closed subspace $U\subset V_ \pi$ that is stable under π, one has U = 0 or $U=V_\pi$.

Example 6.1.6.

Let ${\rm U}(n)$ denote the group of unitary $n\times n$ matrices, so the group of all $u\in{\mathrm M}_n({\mathbb C})$ such that $uu^*=I$ (unit matrix), where $u^*=\bar u^t$. The natural representation of ${\rm U}(n)$ on ${\mathbb C}^n$ is irreducible (See Exercise 6.5).

Definition

Let $(\pi,V_\pi)$ be a representation of G. A vector $v\in V_\pi$ is called a cyclic vector if the linear span of the set $\{\pi(x)v: x\in G\}$ is dense in $V_\pi$. In other words, v is cyclic if the only subrepresentation containing v is the whole of π. It follows that a representation is irreducible if and only if every nonzero vector is cyclic.

Lemma 6.1.7

(Schur) Let $(\pi,V_\pi)$ be a unitary representation of the topological group G. Then the following are equivalent

• (a) $(\pi,V_\pi)$ is irreducible.

• (b) If T is a bounded operator on $V_\pi$ such that $T\pi(g)=\pi(g)T$ for every $g\in G$, then $T\in{\mathbb C}\,{\rm Id}$.

Proof

Since $\pi(g^{-1})=\pi(g)^*$, the set $\{\pi(g):g\in G\}$ is a self-adjoint subset of ${\cal B}(V_\pi)$. Thus the result follows from Theorem 5.1.6. □

Let $(\pi, V_\pi),(\eta,V_\eta)$ be representations of G. A continuous linear operator $T: V_\pi\to V_\eta$ is called a $G$ -homomorphism or intertwining operator if

$$T\pi(g)=\eta(g)T$$

holds for every $g\in G$. We write ${\rm Hom}_G(V_\pi,V_\eta)$ for the set of all G-homomorphisms from $V_\pi$ to $V_\eta$. A nice way to rephrase the Lemma of Schur is to say that a unitary representation $(\pi,V_\pi)$ is irreducible if and only if ${\rm Hom}_G(V_\pi,V_\pi)={\mathbb C}\,{\rm Id}.$

Definition

If $\pi,\eta$ are unitary, they are called unitarily equivalent if there exists a unitary G-homomorphism $T:V_\pi\to V_\eta$.

Example 6.1.8.

Let $G={\mathbb R}$, and let $V_\pi=V_\eta= L^2({\mathbb R})$. The representation π is given by $\pi(x)\phi(y)=\phi(x+y)$ and η is given by $\eta(x)\phi(y)=e^{2\pi\! ixy}\phi(y)$. By Theorem 3.3.1 in [Dei05] (see also Exercise 6.4), the Fourier transform $L^2({\mathbb R})\to L^2({\mathbb R})$ is an intertwining operator from π to η.

Corollary 6.1.9

Let $(\pi,V_\pi)$ and $(\eta,V_\eta)$ be two irreducible unitary representations. Then a G-homomorphism T from $V_\pi$ to $V_\eta$ is either zero or invertible with continuous inverse. In the latter case there exists a scalar $c>0$ such that cT is unitary. The space ${\rm Hom}_G(V_\pi,V_\eta)$ is zero unless π and η are unitarily equivalent, in which case the space is of dimension 1.

Proof

Let $T:V_\pi\to V_\eta$ be a G-homomorphism. Its adjoint $T^*:V_\eta\to V_\pi$ is also a G-homomorphism as is seen by the following calculation for $v\in V_\pi$, $w\in V_\eta$, and $g\in G$,

$$\begin{aligned} {\left\langle{v,T^*\eta(g) w}\right\rangle}&= {\left\langle{Tv,\eta(g)w}\right\rangle}= {\left\langle{\eta(g^{-1})Tv,w}\right\rangle}\\ &= {\left\langle{T\pi(g^{-1})v,w}\right\rangle}= {\left\langle{\pi(g^{-1})v,T^*w}\right\rangle}\\ &= {\left\langle{v,\pi(g)T^*w}\right\rangle}.\end{aligned}$$

This implies that $T^*T$ is a G-homomorphism on $V_\pi$, and therefore it is a multiple of the identity ${\lambda}\,{\rm Id}$ by the Lemma of Schur. If T is non-zero, $T^*T$ is non-zero and positive semi-definite, so ${\lambda}> 0$. Let $c=\sqrt{{\lambda}^{-1}}$, then $(cT)^*(cT)={\rm Id}$. A similar argument shows that $TT^*$ is bijective, which then implies that cT is bijective, hence unitary. The rest is clear. □

Definition

For a locally compact group G we denote by $\widehat{G}$ the set¹ of all equivalence classes of irreducible unitary representations of G. We call $\widehat{G}$ the unitary dual of G. It is quite common to make no notational difference between a given irreducible representation π and its unitary equivalence class $[\pi]$, and we will often do so in this book.

Example 6.1.10

If G is a locally compact abelian group, then every irreducible representation is one-dimensional, and therefore the unitary dual $\widehat{G}$ coincides with the Pontryagin dual of G. To see this, let $(\pi, V_\pi)$ be any irreducible representation of G. Then $\pi(x)\pi(y)=\pi(xy)=\pi(yx)=\pi(y)\pi(x)$ for all $x,y\in G$, so it follows from Schur’s Lemma that $\pi(x)=\lambda(x){\rm Id}_{V_\pi}$ for some $\lambda(x)\in{\mathbb T}$. But this implies that every non-zero closed subspace of $V_\pi$ is invariant, hence must be equal to $V_\pi$. This implies ${\rm dim} V_\pi=1$.

6.2 Representations of $L^1(G)$

A unitary representation $(\pi,V_\pi)$ of G induces an algebra homomorphism from the convolution algebra $L^1(G)$ to the algebra ${\cal B}(V_ \pi)$, as the following proposition shows.

Proposition 6.2.1

Let $(\pi,V_\pi)$ be a unitary representation of the locally compact group G. For every $f\in L^1(G)$ there exists a unique bounded operator $\pi(f)$ on $V_\pi$ such that

$$\left\langle{\pi(f)v,w}\right\rangle=\int_Gf(x){\left\langle{\pi(x)v,w}\right\rangle}\,dx$$

holds for any two vectors $v,w\in V_\pi.$ The induced map $\pi: L^1(G)\to{\cal B}(V_\pi)$ is a continuous homomorphism of Banach-*-algebras.

Proof

Taking complex conjugates one sees that the claimed equation is equivalent to the equality ${\left\langle{w,\pi(f)v}\right\rangle}=\int_G{\overline{f(x)}}{\left\langle{w,\pi(x)v}\right\rangle}\,dx.$ The map $w\mapsto \int_G{\overline{f(x)}}{\left\langle{w,\pi(x)v}\right\rangle}\,dx$ is linear. It is also bounded, since

$$\begin{aligned} \left|\int_G{\overline{f(x)}}{\left\langle{w,\pi(x)v}\right\rangle}\,dx\right| &\le \int_G|f(x){\left\langle{w,\pi(x)v}\right\rangle}|\,dx\\ &\le \int_G|f(x)|\Vert{w}\Vert \Vert{\pi(x)v}\Vert\,dx\\ &= \Vert{f}\Vert_{1}\Vert w\Vert v \Vert.\end{aligned}$$

Therefore, by Proposition C.3.1, there exists a unique vector $\pi(f)v\in V_\pi$ such that the equality holds. It is easy to see that the map $v\mapsto \pi(f)v$ is linear. To see that it is bounded, note that the above shows $\Vert{\pi(f)v}\Vert^2={\left\langle{\pi(f)v,\pi(f)v}\right\rangle}\ \le\ \Vert{f}\Vert_1\Vert v\Vert \Vert{\pi(f) v}\Vert,$ and hence $\Vert{\pi(f)v}\Vert\le\Vert f\Vert_1\Vert v\Vert$. A straightforward computation finally shows $\pi(f*g)=\pi(f)\pi(g)$ and $\pi(f)^*=\pi(f^*)$ for $f,g\in L^1(G)$. □

Alternatively, one can define $\pi(f)$ as the Bochner integral $\pi(f)=\int_G f(x)\pi(x)\,dx$ in the Banach space ${\cal B}(V_\pi)$. By the uniqueness in the above proposition, these two definitions agree.

The above proposition has a converse, as we shall see in Proposition 6.2.3 below.

Lemma 6.2.2

Let $(\pi,V_\pi)$ be a representation of G. Then for every $v\in V_\pi$ and every ${\varepsilon}>0$ there exists a unit-neighborhood U such that for every Dirac function φ _U with support in U one has $\Vert{\pi(\phi_U)v-v}\Vert<{\varepsilon}$. In particular, for every Dirac net $(\phi_U)_U$ the net $(\pi(\phi_U)v)$ converges to v in the norm topology.

Proof

The norm $\Vert{\pi(\phi_U)v-v}\Vert$ equals $\Vert{\int_G\phi_U(x)(\pi(x)v-v)\,dx}\Vert$ and is therefore less than or equal to $\int_G \phi_U(x)\Vert{\pi(x)v-v}\Vert\,dx.$ For given ${\varepsilon}>0$ there exists a unit-neighborhood U ₀ in G such that for $x\in U_0$ one has $\Vert{\pi(x)v-v}\Vert<{\varepsilon}$. For $U\subset U_0$ it follows $\Vert{\pi(\phi_U)v-v}\Vert<{\varepsilon}$. □

Definition

We say that a $*$-representation $\pi:L^1(G)\to{\cal B}(V)$ of $L^1(G)$ on a Hilbert space V is non-degenerate if the vector space

$\pi ({{L}^{1}}(G))V\ \overset{\text{def}}{\mathop{=}}\,\text{span}\{\pi (f)v:f\in {{L}^{1}}(G),v\in V\}$

is dense in V. It follows from the above lemma that every representation of $L^1(G)$ that comes from a representation $(\pi, V_\pi)$ as in Proposition 6.2.1, is non-degenerate. The next proposition gives a converse to this.

Proposition 6.2.3

Let $\pi:L^1(G)\to{\cal B}(V)$ be a non-degenerate $*$- representation on a Hilbert space V. Then there exists a unique unitary representation $(\tilde\pi, V)$ of G such that ${\left\langle{\pi(f)v,w}\right\rangle}=\int_G f(x){\left\langle{\tilde\pi(x)v,w}\right\rangle}\,dx$ holds for all $f\in L^1(G)$ and all $v,w\in V$.

Proof

Note first that π is continuous by Lemma 2.7.1. We want to define an operator $\tilde\pi(x)$ on the dense subspace $\pi(L^1(G))V$ of V. This space is made up of sums of the form $\sum_{i=1}^n\pi(f_i)v_i$ for $f_i\in L^1(G)$ and $v_i\in V$. We propose to define $\tilde{\pi }(x)\sum\limits_{i=1}^{n}{\pi }({{f}_{i}}){{v}_{i}}\ \overset{\text{def}}{\mathop{=}}\,\sum\limits_{i=1}^{n}{\pi }({{L}_{x}}{{f}_{i}}){{v}_{i}}.$ We have to show well-definedness, which amounts to show that if $\sum_{i=1}^n\pi(f_i)v_i=0$, then $\sum_{i=1}^n\pi(L_xf_i)v_i=0$ for every $x\in G$. For $x\in G$ and $f\!,g\in L^1(G)$ a short computation shows that $g^**(L_xf)=(L_{x^{-1}}g)^**f$. Based on this, we compute for $v,w\in V$ and $f_1,\dots, f_n\in L^1(G)$,

$$\begin{aligned} {\left\langle{\sum_{i=1}^n\pi(L_xf_i)v,\pi(g)w}\right\rangle} &=\sum_{i=1}^n{\left\langle{\pi\big(g^**(L_xf_i)\big)v,w}\right\rangle}\\ = \sum_{i=1}^n{\left\langle{\pi((L_{x^{-1}}g)^**f_i)v,w}\right\rangle}&={\left\langle{\sum_{i=1}^n\pi(f_i)v, \pi(L_{x^{-1}}g)w}\right\rangle}.\end{aligned}$$

Now for the well-definedness of $\tilde\pi(x)$ assume $\sum_{i=1}^n\pi(f_i)v_i=0$, then the above computation shows that the vector $\sum_{i=1}^n\pi(L_xf_i)v_i$ is orthogonal to all vectors of the form $\pi(g)w$, which span the dense subspace $\pi(L^1(G))V$, hence $\sum_{i=1}^n\pi(L_xf_i)v_i=0$ follows. The computation also shows that this, now well-defined operator $\tilde\pi(x)$ is unitary on the space $\pi(L^1(G))V$ and since the latter is dense in V, the operator $\tilde\pi(x)$ extends to a unique unitary operator on V with inverse $\tilde\pi(x^{-1})$, and we clearly have $\tilde\pi(xy)=\tilde\pi(x)\tilde\pi(y)$ for all $x,y\in G$. Since for each $f\in L^1(G)$ the map $G\to L^1(G)$ sending x to L _xf is continuous by Lemma 1.4.2, it follows that $x\mapsto \tilde\pi(x)v$ is continuous for every $v\in V$. Thus $(\tilde\pi,V)$ is a unitary representation of G.

It remains to show that $\pi(f)$ equals $\tilde\pi(f)$ for every $f\in L^1(G)$. By continuity it is enough to show that ${\left\langle{\tilde\pi(f)\pi(g)v,w}\right\rangle}={\left\langle{\pi(f)\pi(g)v,w}\right\rangle}$ for all $f\!,g\in C_c(G)$ and $v,w\in V$. Since $g\mapsto{\left\langle{\pi(g)v,w}\right\rangle}$ is a continuous linear functional on $L^1(G)$ we can use Lemma B.6.5 to get

$$\begin{aligned} {\left\langle{\tilde\pi(f)\pi(g)v,w}\right\rangle}&= \int_G f(x){\left\langle{\tilde\pi(x)(\pi(g)v),w}\right\rangle}\,dx\\ &= \int_G {\left\langle{\pi(f(x)L_xg)v,w}\right\rangle}\,dx\\ &={\left\langle{\pi\left(\int_G f(x)L_xg\,dx\right)v,w}\right\rangle}\\ &={\left\langle{\pi(f*g)v,w}\right\rangle}={\left\langle{\pi(f)\pi(g)v,w}\right\rangle},\end{aligned}$$

which completes the proof. □

Remark 6.2.4

If we define unitary equivalence and irreducibility for representations of $L^1(G)$ in the same way as we did for unitary representations of G, then it is easy to see that the one-to-one correspondence between unitary representations of G and non-degenerate $*$-representations of $L^1(G)$ preserves unitary equivalence and irreducibility in both directions. Note that an irreducible representation π of $L^1(G)$ is automatically non-degenerate, since the closure of $\pi(L^1(G))V_\pi$ is an invariant subspace of $V_\pi$. Thus, we obtain a bijection between the space $\widehat{G}$ of equivalence classes of irreducible representations of G and the set $L^1(G){\widehat{}}$ of irreducible $*$-representations of $L^1(G)$.

Example 6.2.5

Consider the left regular representation on G. Then the corresponding representation $L: L^1(G)\to{\cal B}(L^2(G))$ is given by the convolution operators $L(f)\phi=f*\phi$ whenever the convolution $f*\phi$ makes sense.

6.3 Exercises

Exercise 6.1

Let G be a topological group and let V be a Banach space. We equip the group ${\rm GL}_{\rm cont}(V)$ with the topology induced by the operator norm. Show that any continuous group homomorphism $G\to{\rm GL}_{\rm cont}(V)$ is a representation but that not every representation is of this form.

Exercise 6.2

If π is a unitary representation of the locally compact group G, then $\Vert{\pi(g)}\Vert=1$. Give an example of a representation π, for which the map $g\mapsto\Vert{\pi(g)}\Vert$ is not bounded on G.

Exercise 6.3

Let I be an index set, and for $i\in I$ let $(\pi_i,V_i)$ be a unitary representation of the locally compact group G. Let $V=\bigoplus_{i\in I}V_i$ be the Hilbert direct sum (See Appendix C.3). Define the map $\pi:G\to{\cal B}(V)$ by

$$\pi(g)\sum_i v_i=\sum_i\pi_i(g)v_i.$$

Show that this is a unitary representation of the group G. It is called the direct sum representation.

Exercise 6.4

Show that the Fourier transform on ${\mathbb R}$ induces a unitary equivalence between the the unitary representations π and η of ${\mathbb R}$ on $L^2({\mathbb R})$ given by $\pi(x)\phi(y)=\phi(x+y)$ and $\eta(x)\phi(y)=e^{2\pi\! i xy}\phi(y)$.

Exercise 6.5

Show that the natural representation of ${\rm U}(n)$ on ${\mathbb C}^n$ is irreducible.

Exercise 6.6

(a) For $t\in{\mathbb R}$ let $A(t)=\left(\begin{smallmatrix}1 & t \\ 0 & 1\end{smallmatrix} \right)$. Show that A(t) is not conjugate to a unitary matrix for $t\ne 0$.

(b) Let P be the group of upper triangular matrices in ${\rm SL}_2({\mathbb R})$. The injection $\eta:P\hookrightarrow{\rm GL}_2({\mathbb C})$ can be viewed as a representation on $V={\mathbb C}^2$. Show that η is not the sum of irreducible representations. Determine all irreducible subrepresentations.

Exercise 6.7

Let G be a locally compact group and H a closed subgroup. Let $(\pi,V_\pi)$ be an irreducible unitary representation of G, and let

$$V_\pi^H=\{v\in V_\pi: \pi(h)v=v\ \forall h\in H \}$$

be the space of H-fixed vectors. Show: If H is normal in G, then $V_\pi^H$ is either zero or the whole space $V_\pi$.

Exercise 6.8

Show that $G={\rm SL}_2({\mathbb R})$ has no finite dimensional unitary representations except the trivial one.

Instructions:

• For $m\in{\mathbb N}$ show

  $$\left(\begin{array}{cc}m & \\ & m^{-1}\end{array}\right) A(t) \left(\begin{array}{cc}m & \\ & m^{-1}\end{array}\right)^{-1} = A(m^2t)=A (t)^{m^2}.$$

  Let $\phi: G\to{\rm U}(n)$ be a representation. Show that the eigenvalues of $\phi(A(t))$ are a permutation of their m-th powers for every $m\in{\mathbb N}$. Conclude that they all must be equal to 1.

• Show that the normal subgroup of G generated by $\{A(t): t\in{\mathbb R}\}$ is the whole group.

Exercise 6.9

Let $(\pi,V_\pi)$ be a unitary representation of the locally compact group G. Let $f\in L^1(G)$. Show that the Bochner integral

$$\int_G f(x)\pi(x)\,dx\ \in\ {\cal B}(V_\pi)$$

exists and that the so defined operator coincides with $\pi(f)$ as defined in Proposition 6.2.1.

(Hint: Use Corollary 1.3.6 (d) and Lemma B.6.2 as well as Proposition B.6.3.)

Exercise 6.10

In Lemma 6.2.2 we have shown that for a representation π and Dirac functions φ _U the numbers $\Vert{\pi(\phi_U)v-v}\Vert$ become arbitrarily small for fixed $v\in V_ \pi$. Give an example, in which $\Vert{\pi(\phi_U)-{\rm Id}}\Vert_{\rm op}$ does not become small as the support of the Dirac function φ _U shrinks.

Exercise 6.11

Give an example of a representation that possesses cyclic vectors without being irreducible.

6.3.1 Notes

As for abelian groups, one can associate to each locally compact group G the group $C^*$-algebra $C^*(G)$. It is defined as the completion of $L^1(G)$ with respect to the norm

$\|f{{\|}_{{{C}^{*}}}}\ \overset{\text{def}}{\mathop{=}}\,\sup \{\|\pi (f)\|:\pi \ \text{a}\,\text{unitary}\,\text{representation}\,\text{of}\,\text{G}\},$

which is finite since $\|\pi(f)\|\leq \|f\|_1$ for every unitary representation π of G. By definition of the norm, every unitary representation π of G extends to a $*$-representation of $C^*(G)$, and, as for $L^1(G)$, this extension provides a one-to-one correspondence between the unitary representation of G to the non-degenerate $*$-representations of $C^*(G)$. Therefore, the rich representation theory of general $C^*$-algebras, as explained beautifully in Dixmier’s classic book [Dix96] can be used for the study of unitary representations of G. For a more recent treatment of $C^*$-algebras related to locally compact groups we also refer to Dana William’s book [Wil07].