§ 2. Linear representations of groups

1. Continuous linear representations

Let G be a topological group, E a locally convex space, U a linear representation of G in E.

Definition 1

— (i) $U$ is said to be separately continuous if, for every $s \in G$ , $U(s)$ is a continuous endomorphism of $E$ , and if, for every $x \in E$ , the mapping $s \mapsto U(s)x$ of $G$ into $E$ is continuous.

(ii) $U$ is said to be continuous if $(s, x) \mapsto U(s)x$ is a continuous mapping of $G \times E$ into $E$ .

(iii) $U$ is said to be equicontinuous if it is continuous and if the set of endomorphisms $U(s)$ , where $s$ runs over $G$ , is equicontinuous.

Remark 1

To say that $U$ is separately continuous means that $s \mapsto U(s)$ is a continuous mapping of $G$ into the space $\mathscr{L}(E; E)$ of continuous endomorphisms of $E$ , equipped with the topology of pointwise convergence.

Remark 2

To say that $U$ is continuous is equivalent to the following set of three conditions:

a) for every $s \in G$ , $U(s)$ is continuous; b) there exists a neighborhood $V$ of $e$ such that $U(V)$ is equicontinuous; c) there exists a total set $D$ in $E$ such that, for every $x \in D$ , the mapping $s \mapsto U(s)x$ is continuous.

These conditions are obviously necessary. Conversely, suppose that the conditions a), b), c) are satisfied. On $U(V)$ , the topology of pointwise convergence is identical to the topology of pointwise convergence in $D$ (TVS, III, §3, No. 4, Prop. 5). Therefore the mapping $(s, x) \mapsto U(s)x$ of $V \times E$ into $E$ is continuous (GT, X, §2, No. 1, Cor. 3 of Prop. 1). Since $U(s_0s)x = U(s_0)(U(s)x)$ for all $s_0 \in G$ , $s \in G$ , $x \in E$ , one sees that $U$ is continuous.

When $G$ is locally compact, the conditions a) and b) are equivalent to the condition: $a')$ for every compact subset $K$ of $G$ , $U(K)$ is equicontinuous.

Remark 3

Suppose that $U$ is a continuous linear representation of $G$ in $E$ . For every $s \in G$ , let $\widehat{U}(s)$ be the continuous extension of $U(s)$ to the completion $\widehat{E}$ of $E$ . Then $\widehat{U}$ is a linear representation of $G$ in $\widehat{E}$ , satisfying conditions a) and c) of Remark 2, and also condition b) by GT, X, §2, No. 2, Prop. 4. Therefore $\widehat{U}$ is a continuous linear representation of $G$ in $\widehat{E}$ .

Remark 4

When $E$ is a normed space, $U$ is said to be isometric if $\|U(s)\| = 1$ for every $s \in G$ . For this, it suffices that $\|U(s)\| \leq 1$ for all $s \in G$ , because one then has

1 = \|1\| \leq \|U(s)\| \cdot \|U(s^{-1})\|,

whence

\|U(s)\| = \|U(s^{-1})\| = 1

for all

s \in G

Proposition 1

— If $G$ is a locally compact group and $E$ is barreled, then every separately continuous linear representation $U$ of $G$ in $E$ is continuous.

For every compact subset $K$ of $G$ , $U(K)$ is compact for the topology of pointwise convergence (Remark 1), therefore is equicontinuous (TVS, III, §4, No. 2, Th. 1); one then applies Remark 2.

Lemma 1

Let G be a locally compact group, $\rho$ a lower semi-continuous finite numerical function $\geqslant 0$ on G such that $\rho(st) \leqslant \rho(s)\rho(t)$ for all $s,t \in G$ . Then $\rho$ is bounded above on every compact subset of G.

There exists a nonempty open subset U of G such that $\rho$ is bounded above on U (GT, IX, §5, No. 4, Th. 2). Let K be a compact subset of G. Then K is covered by a finite number of sets $s_1U, \ldots, s_nU$ . For every $x \in U$ , one has $\rho(s_ix) \leqslant \rho(s_i)\rho(x)$ , therefore $\rho$ is bounded above on the $s_iU$ , hence on K.

Lemma 2

Let G be a topological group, U a linear representation of G in a normed space E, and A a dense subset of E. Assume that for every $s \in G$ , $U(s)$ is continuous, and that, for every $x \in A$ , $s \mapsto U(s)x$ is a continuous mapping of G into E. Then the function $s \mapsto g(s) = \|U(s)\|$ on G is lower semi-continuous and satisfies $g(st) \leqslant g(s)g(t)$ .

Let B be the unit ball of E. Then $g(s) = \sup_{x \in B \cap A} \|U(s)x\|$ , and each function $s \mapsto \|U(s)x\|$ is continuous on G, therefore $g$ is lower semi-continuous. On the other hand,

g(st) = \|U(s)U(t)\| \leqslant \|U(s)\| \cdot \|U(t)\| = g(s)g(t).

Proposition 2

Let G be a locally compact group, U a linear representation of G in a normed space E. Let A be a dense subset of E. Assume that for every $s \in G$ , $U(s)$ is continuous and that, for every $x \in A$ , $s \mapsto U(s)x$ is a continuous mapping of G into E. Then U is continuous.

For, $\|U(s)\|$ is bounded on every compact subset of G by Lemmas 1 and 2, and one then applies Remark 2.

2. Contragredient representation

Let U be a separately continuous linear representation of G in E. Let E' be the dual of E. The mapping $s \mapsto {}^tU(s)$ is a linear representation in E' of the group $G^0$ opposite G; we shall say that this representation is the transpose of U. The mapping $s \mapsto {}^tU(s^{-1}) = {}^tU(s)^{-1}$ is a linear representation of G in E', called the contragredient of U.

Lemma 3

Let X be a locally compact space, Y and Z topological spaces, $\varphi$ a continuous mapping of $X \times Y$ into Z, and $\varphi_x$ the mapping $y \mapsto \varphi(x,y)$ of Y into Z. The spaces $\mathscr{C}(Y), \mathscr{C}(Z)$ being equipped with the topology of compact convergence, the mapping $(x,f) \mapsto f \circ \varphi_x$ of $X \times \mathscr{C}(Z)$ into $\mathscr{C}(Y)$ is continuous.

It clearly suffices to consider the case that X is compact. Let $(x_0, f_0) \in X \times \mathscr{C}(Z)$ , K a compact subset of Y, and $\varepsilon > 0$ . Let $K' = \varphi(X \times K)$ . Since $f_0 \circ \varphi$ is uniformly continuous in $X \times K$ , there exists a neighborhood W of $x_0$ such that $|f_0(\varphi(x,y)) - f_0(\varphi(x_0,y))| \leq \varepsilon$ for $x \in W$ and $y \in K$ . On the other hand, if one takes $f \in \mathscr{C}(Z)$ to be such that $|f(z) - f_0(z)| \leq \varepsilon$ for all $z \in K'$ , one will have $|f(\varphi(x,y)) - f_0(\varphi(x,y))| \leq \varepsilon$ for $x \in X, y \in K$ , and therefore $|f(\varphi(x,y)) - f_0(\varphi(x_0,y))| \leq 2\varepsilon$ for $x \in W, y \in K$ . Whence the lemma.

Let us now return to the earlier notations.

Proposition 3

(i) If U is separately continuous, then $^tU$ is separately continuous when E' is equipped with the weak topology $\sigma(E', E)$ . (ii) *If G is locally compact and U is continuous, then $^tU$ is continuous when E' is equipped with the topology of compact convergence.

The assertion (i) is immediate. The assertion (ii) follows from Lemma 3 where one has taken $X = G, Y = Z = E, \varphi(s, x) = U(s)x$ .

3. Example: linear representations in spaces of continuous functions

Let G be a discrete group operating on the left on a set X. A complex function $\chi$ on $G \times X$ is called a multiplier if

(1) $\chi(e, x) = 1$ for all $x \in X;$ (2) $\chi(st, x) = \chi(s, tx)\chi(t, x)$ for all $s, t$ in G, $x \in X$ .

It follows that

(3) $\chi(t^{-1}, tx)\chi(t, x) = 1$ for all $t \in G, x \in X,$

and in particular $\chi(t, x) \neq 0$ for all $t \in G, x \in X$ .

For every complex function $f$ defined on X and every $s \in G$ , let $\gamma_\chi(s)f$ be the complex function on X defined by

(4) $(\gamma_\chi(s)f)(x) = \chi(s^{-1}, x)f(s^{-1}x)$ .

Then $\gamma_\chi(e)f = f$ and

\begin{align*} (\gamma_\chi(s)\gamma_\chi(s')f)(x) &= \chi(s^{-1}, x)(\gamma_\chi(s')f)(s^{-1}x) \\ &= \chi(s^{-1}, x)\chi({s'}^{-1}, s^{-1}x)f({s'}^{-1}s^{-1}x) \\ &= \chi((ss')^{-1}, x))f((ss')^{-1}x) = (\gamma_\chi(ss')f)(x), \end{align*}

thus $\gamma_\chi$ is a linear representation of G. For $\chi = 1$ , one recovers the endomorphisms $\gamma(s)$ (Ch. VII, §1, No. 1, formula (3)).

Suppose now that G and X are locally compact, G operating continuously on X, and $\chi$ continuous on $G \times X$ . Then $C(X)$ and $K(X)$ are stable for the $\gamma_\chi(s)$ , whence linear representations of G in $C(X)$ and $K(X)$ which we shall again denote $\gamma_\chi$ .

Proposition 4

The linear representations $\gamma_\chi$ of G in $C(X)$ and $K(X)$ are continuous.

The mapping $(s, f) \mapsto (s, \gamma(s)f)$ of $G \times C(X)$ into $G \times C(X)$ is continuous (No. 2, Lemma 3). On the other hand, the mapping $(s, f) \mapsto \chi(s, \cdot)f$ of $G \times C(X)$ into $C(X)$ is continuous; for, if s tends to $s_0$ in G, then $\chi(s, \cdot)$ tends to $\chi(s_0, \cdot)$ uniformly on every compact subset of X; if, moreover, $f$ tends to $f_0$ in $C(X)$ , then $\chi(s, \cdot)f$ tends to $\chi(s_0, \cdot)f_0$ uniformly on every compact subset of X, whence our assertion. Thus the representation $\gamma_\chi$ of G in $C(X)$ is continuous.

Let us show that the representation $\gamma_\chi$ of G in $K(X)$ is continuous. Since $K(X)$ is the direct limit of Banach spaces, it is barreled (TVS, III, §4, No. 1, Cor. 3 of Prop. 3), thus it suffices to prove that $\gamma_\chi$ is separately continuous (No. 1, Prop. 1). Now, let H be a compact subset of X and let $s_0 \in G$ . Let V be a compact neighborhood of $s_0$ in G, and let L = VH, which is compact in X. For every $f \in K(X, H)$ , the support of $\gamma_\chi(s_0)f$ is contained in L, and

\sup_{x \in X} |(\gamma_\chi(s_0)f)(x)| \leq \sup_{x \in L} |\chi(s_0^{-1}, x)| \cdot \sup_{x \in X} |f(x)|,

therefore

f \mapsto \gamma_\chi(s_0)f

is a continuous linear mapping of

K(X, H)

into

K(X, L)

; it follows that

f \mapsto \gamma_\chi(s_0)f

is a continuous linear mapping of

K(X)

into itself (TVS, II, §4, No. 4, Prop. 5). On the other hand, the topology of

K(X, L)

is induced by that of

C(X)

. By what has already been proved, the mapping

s \mapsto \gamma_\chi(s)f

of V into

K(X, L)

is continuous. This completes the proof that

\gamma_\chi

is separately continuous.

Proposition 5

Suppose that each function $\chi(s, \cdot)$ is bounded. Then $\gamma_\chi$ leaves $\overline{K(X)}$ stable, and the linear representation $\gamma_\chi$ of G in $\overline{K(X)}$ is continuous.

It is clear that $\gamma_\chi$ leaves $\overline{K(X)}$ stable and that each of the $\gamma_\chi(s)$ is continuous in $\overline{K(X)}$ . On the other hand, for every $f \in K(X)$ , $s \mapsto \gamma_\chi(s)f$ is a continuous mapping of G into $K(X)$ and a fortiori into $\overline{K(X)}$ . Therefore the representation $\gamma_\chi$ in $\overline{K(X)}$ is continuous (No. 1, Prop. 2).

4. Example: linear representations in spaces of measures

Again let G be a locally compact group, operating continuously on the left in a locally compact space X, and let $\chi$ be a continuous multiplier on $G \times X$ . The linear representation $\gamma_\chi$ of $G$ in $\mathscr{K}(X)$ admits a contragredient representation in $\mathscr{M}(X)$ , which we shall again denote by $\gamma_\chi$ , and which is defined by the following formula (where $\mu \in \mathscr{M}(X) , f \in \mathscr{K}(X)$ ):

\langle \gamma_\chi(s)\mu, f \rangle = \langle \mu, \gamma_\chi(s^{-1})f \rangle = \langle \chi(s, \cdot) \cdot \mu, \gamma(s^{-1})f \rangle = \langle \gamma(s)(\chi(s, \cdot) \cdot \mu), f \rangle ,

whence

\gamma_\chi(s)\mu = \gamma(s)(\chi(s, \cdot) \cdot \mu) = (\gamma(s)\chi(s, \cdot)) \cdot (\gamma(s)\mu) .

We note that

(\gamma(s)\chi(s, \cdot))(x) = \chi(s, s^{-1}x) .

The linear representation $\gamma_\chi$ of $G$ in $\mathscr{C}(X)$ admits a contragredient representation in the space $\mathscr{C}'(X)$ of measures on $X$ with compact support, a representation which we again denote by $\gamma_\chi$ ; the endomorphisms $\gamma_\chi(s)$ of $\mathscr{C}'(X)$ are the restrictions of the endomorphisms $\gamma_\chi(s)$ of $\mathscr{M}(X)$ .

Proposition 6

If one equips $\mathscr{M}(X)$ (resp. $\mathscr{C}'(X)$ ) with the topology of uniform convergence in the compact subsets of $\mathscr{K}(X)$ (resp. $\mathscr{C}(X)$ ), then the linear representation $\gamma_\chi$ of $G$ in $\mathscr{M}(X)$ (resp. $\mathscr{C}'(X)$ ) is continuous.

Proposition 7

Suppose that each function $\chi(s, \cdot)$ is bounded. Then $\gamma_\chi$ leaves stable $\mathscr{M}^1(X)$ and, if $\mathscr{M}^1(X)$ is equipped with the topology of uniform convergence in the compact subsets of $\mathscr{K}(X)$ , then the linear representation $\gamma_\chi$ of $G$ in $\mathscr{M}^1(X)$ is continuous.* These propositions result from Props. 3, 4, 5.

5. Example: linear representations in the spaces $L^p$

Again let $G$ be a locally compact group, operating continuously on the left in a locally compact space $X$ . Let $\beta$ be a positive measure on $X$ with support $X$ . Let us assume that there exists a continuous function $\chi > 0$ on $G \times X$ such that, for every $s \in G$ ,

\gamma(s)\beta = \chi(s^{-1}, \cdot) \cdot \beta

(which implies in particular that $\beta$ is quasi-invariant under $G$ ). Then, $\chi$ is a multiplier. For, given $s, t$ in $G$ , one has

\gamma(s)\gamma(t)\beta = \gamma(s)(\chi(t^{-1}, \cdot) \cdot \beta) = (\gamma(s)\chi(t^{-1}, \cdot)) \cdot (\gamma(s)\beta) = (\gamma(s)\chi(t^{-1}, \cdot)) \cdot \chi(s^{-1}, \cdot) \cdot \beta ,

\gamma(st)\beta = \chi(t^{-1}s^{-1}, \cdot) \cdot \beta ,

therefore

\chi(t^{-1}, s^{-1}x)\chi(s^{-1}, x) = \chi(t^{-1}s^{-1}, x)

locally

\beta

-almost everywhere, consequently everywhere, since

\chi

is continuous and

\beta

has support

X

Let $p \in [1, +\infty[$ . For every $f \in \mathscr{L}_C^p(X, \beta)$ and every $s \in G$ , let $\gamma_{\chi,p}(s)f$ be the function on $X$ defined by

(\gamma_{\chi,p}(s)f)(x) = \chi(s^{-1}, x)^{1/p} f(s^{-1}x).

One has

\int^* |\chi(s^{-1}, x)^{1/p} f(s^{-1}x)|^p d\beta(x) = \int^* |f(s^{-1}x)|^p \chi(s^{-1}, x) d\beta(x)

= \int |f(x)|^p d\beta(x),

therefore

\gamma_{\chi,p}(s)f \in \mathscr{L}_C^p(X, \beta)

. One sees that

\gamma_{\chi,p}(s)

is an isometric endomorphism of

\mathscr{L}_C^p(X, \beta)

and defines, by passage to the quotient, an isometric endomorphism of

L_C^p(X, \beta)

, also denoted

\gamma_{\chi,p}(s)

. On the other hand,

\chi^{1/p}

is obviously a multiplier, therefore

\gamma_{\chi,p}

is a linear representation of

G

L_C^p(X, \beta)

by what we have seen in No. 3.

Proposition 8

The linear representation $\gamma_{\chi,p}$ of $G$ in $L_C^p(X, \beta)$ is continuous and isometric.

Let $f \in \mathscr{K}(X)$ . When $s$ tends to $s_0$ in $G$ , $\gamma_{\chi,p}(s)f$ tends to $\gamma_{\chi,p}(s_0)f$ in $\mathscr{K}(X)$ , hence in $L_C^p(X, \beta)$ . Since the $\gamma_{\chi,p}(s)$ are isometric, Prop. 8 is obtained by applying Remark 2 of No. 1.

For the case that $\chi$ is not assumed continuous, cf. §4, Exer. 13.

Proposition 9

Suppose that each function $\chi(s, \cdot)$ is bounded. Then $\gamma_\chi$ leaves $L_C^p(X, \beta)$ stable, and the linear representation $\gamma_\chi$ of $G$ in $L_C^p(X, \beta)$ is continuous.

Let $f \in \mathscr{L}_C^p(X, \beta)$ . Then

\int^* |\chi(s^{-1}, x)f(s^{-1}x)|^p d\beta(x)

\leq \sup_{x \in X} \chi(s^{-1}, x)^{p-1} \int^* |f(s^{-1}x)|^p \chi(s^{-1}, x) d\beta(x)

= \sup_{x \in X} \chi(s^{-1}, x)^{p-1} \int |f(x)|^p d\beta(x),

therefore

\gamma_\chi(s)f \in \mathscr{L}_C^p(X, \beta)

, and

\| \gamma_\chi(s) \| \leq \sup_{x \in X} \chi(s^{-1}, x)^{1/q},

(5) where

q

denotes the exponent conjugate to

p

. If

f \in \mathscr{K}(X)

, then

\gamma_{\chi}(s)f

tends to

\gamma_{\chi}(s_0)f

\mathscr{K}(X)

, hence in

\mathscr{L}_C^p(X, \beta)

, as

s

tends to

s_0

. Therefore the representation

\gamma_{\chi}

G

L_C^p(X, \beta)

is continuous (No. 1, Prop. 2).

Properties analogous to those of Nos. 3, 4, 5 hold if $G$ operates on the right in $X$ .

In particular, if one regards $G$ as operating on itself by left or right translations, and if one takes $\chi = 1$ , one obtains the left and right regular representations of $G$ in $\mathscr{C}(G)$ , $\mathscr{K}(G)$ , $\overline{\mathscr{K}}(G)$ , $\mathscr{C}'(G)$ , $\mathscr{M}(G)$ , $\mathscr{M}^1(G)$ . If one takes $\beta$ to be a left (resp. right) Haar measure on $G$ , and if one takes $\chi = 1$ , one obtains the left (resp. right) regular representation of $G$ in $L_C^p(G, \beta)$ .

6. Extension of a linear representation of $G$ to the measures on $G$

Let $G$ be a locally compact group, $E$ a locally convex space, $U$ a linear representation of $G$ in $E$ . Assume $U$ to be continuous and $E$ quasi-complete. Then, for every measure $\mu \in \mathscr{C}'(G)$ , one has

\int_G U(s)\, d\mu(s) \in \mathscr{L}(E; E)

(Ch. VI, §1, No. 7). We shall write $U(\mu) = \int_G U(s)\, d\mu(s)$ . We equip $\mathscr{C}'(G)$ with the topology of compact convergence in $\mathscr{C}(G)$ . The mapping $(\mu, x) \mapsto U(\mu)x$ of $\mathscr{C}'(G) \times E$ into $E$ is hypocontinuous relative to the equicontinuous subsets of $\mathscr{C}'(G)$ and the compact subsets of $E$ ; in particular, the mapping $\mu \mapsto U(\mu)$ of $\mathscr{C}'(G)$ into $\mathscr{L}(E; E)$ (equipped with the topology of compact convergence) is continuous (loc. cit., Prop. 16).

In order to be able to apply these results later on, we note that if $X$ is a locally compact space then $\mathscr{C}(X)$ , equipped with the topology of compact convergence, is complete (GT, X, §1, No. 6, Cor. 3 of Th. 2). On the other hand, $\mathscr{K}(X)$ is barreled, therefore its dual $\mathscr{M}(X)$ , equipped with the topology of compact convergence in $\mathscr{K}(X)$ , is quasi-complete (TVS, III, §4, No. 2, Cor. 4 of Th. 1). Of course, $\mathscr{K}(X)$ is complete for the topology deduced from its norm, therefore its dual $\mathscr{M}^1(X)$ is quasi-complete for the topology of compact convergence in $\mathscr{K}(X)$ (loc. cit.).

Let us now assume that $U$ is a continuous linear representation of the locally compact group $G$ on a Banach space $E$ . Set $g(s) = \|U(s)\|$ for all $s \in G$ . Then, if $\mu$ is a measure on $G$ such that $g$ is $\mu$ -integrable, one has $\int_G U(s)\, d\mu(s) \in \mathscr{L}(E; E)$ and $\|\int_G U(s)\, d\mu(s)\| \leq \int g(s)\, d|\mu|(s)$ (Ch. VI, §1, No. 7, Remark 1). We again write $U(\mu) = \int_G U(s)\, d\mu(s)$ .

7. Relations between the endomorphisms $U(\mu)$ and the endomorphisms $U(s)$

Lemma 4

Let $T$ be a locally compact space, $a$ a point of $T$ , $M$ a subset of $\mathscr{M}(T)$ , and $\mathfrak{F}$ a filter on $M$ . Assume that: (i) for every compact subset $K$ of $T$ , the numbers $|\mu|(K)$ , for $\mu \in M$ , are bounded above; (ii) $\lim_{\mu,\mathfrak{F}} |\mu|(K) = 0$ for every compact subset $K$ of $T - \{a\}$ . (iii) there exists a compact neighborhood $V$ of $a$ in $T$ such that $\lim_{\mu,\mathfrak{F}} \mu(V) = 1$ .

Then the filter $\mathfrak{F}$ converges to $\varepsilon_a$ in $\mathscr{M}(T)$ equipped with the topology of compact convergence in $\mathscr{H}(T)$ .

By the hypothesis (i), $M$ is an equicontinuous subset of $\mathscr{M}(T)$ since it is vaguely bounded and $\mathscr{H}(T)$ is barreled (TVS, III, §4, No. 2, Th. 1). It therefore suffices (GT, X, §2, No. 4, Th. 1) to prove that if $f \in \mathscr{H}(T)$ , then $\lim_{\mu,\mathfrak{F}} \mu(f) = f(a)$ . Let $K$ be the union of $V$ and the support of $f$ ; if $K'$ is the closure of $K - V$ , one has

|\mu(K) - \mu(V)| = |\mu(K - V)| \leq |\mu|(K');

since $K'$ is compact and does not contain $a$ , one concludes from this that $\lim_{\mu,\mathfrak{F}} \mu(K) = 1$ . Let $\varepsilon > 0$ , and let $W$ be an open neighborhood of $a$ in $K$ such that $|f(t) - f(a)| \leq \varepsilon$ for $t \in W$ ; one can write

\mu(f) - f(a) = f(a)(\mu(K) - 1) + \int_K (f(t) - f(a)) d\mu(t);

the integral over $K$ may be written as the sum of the analogous integrals over $W$ and $K - W$ ; if $C = \sup |f|$ , one therefore has

|\mu(f) - f(a)| \leq C|\mu(K) - 1| + \varepsilon \cdot |\mu|(K) + 2C \cdot |\mu|(K - W).

Since the first and third terms on the right side tend to 0 with respect to $\mathfrak{F}$ , one sees that indeed $\lim_{\mu,\mathfrak{F}} \mu(f) = f(a)$ .

Corollary 1

With hypotheses as in Lemma 4, suppose in addition that there exists a compact subset $K_0$ of $T$ containing the supports of all of the measures $\mu \in M$ . Then $\mathfrak{F}$ also converges to $\varepsilon_a$ in $\mathscr{C}'(T)$ equipped with the topology of compact convergence in $\mathscr{C}(T)$ .

For, the restriction mapping of $\mathscr{C}(T)$ into $\mathscr{C}(K_0)$ is continuous; therefore, if $H$ is a compact subset of $\mathscr{C}(T)$ , then the restrictions to $K_0$ of the functions in $H$ form a compact subset of $\mathscr{C}(K_0)$ . It then suffices to apply Lemma 4 on replacing $T$ by $K_0$ .

Corollary 2

With hypotheses as in Cor. 1, let $f$ be a continuous mapping of $T$ into a quasi-complete locally convex space $E$ . Then

\lim_{\mu,\mathfrak{T}} \int f(t)\, d\mu(t) = f(a).

This follows from Cor. 1, and Prop. 14 of Ch. VI, §1, No. 6.

Corollary 3

Let $G$ be a locally compact group, $E$ a quasi-complete locally convex space, and $U$ a continuous linear representation of $G$ in $E$ . Let $\beta$ be a positive measure on $G$ , $a$ an element of $G$ , and $\mathscr{B}$ a base for the filter of neighborhoods of $a$ , formed of compact neighborhoods. For every $V \in \mathscr{B}$ , let $f_V$ be a continuous function $\geqslant 0$ on $G$ , with support contained in $V$ , and such that $\int f_V\, d\beta = 1$ . Then, for every $x \in E$ ,

U(a)x = \lim_V U(f_V \cdot \beta)x,

the limit being taken with respect to the section filter of $\mathscr{B}$ .

The mapping $s \mapsto U(s)x$ of $G$ into $E$ is continuous. By Cor. 2, $U(a)x = \lim_V \int (U(s)x) \cdot f_V(s)\, d\beta(s)$ with respect to the section filter of $\mathscr{B}$ , that is, $U(a)x = \lim_V U(f_V \cdot \beta)x$ .

Proposition 10

Let $G$ be a locally compact group, $E$ a quasi-complete locally convex space, $U$ a continuous linear representation of $G$ in $E$ , and $\beta$ a positive measure on $G$ with support $G$ .

(i) The vectors $U(f \cdot \beta)x$ , where $f$ runs over $\mathscr{K}(G)$ and $x$ runs over $E$ , are dense in $E$ .

(ii) *Let $F$ be a closed linear subspace of $E$ . If $F$ is stable for $U$ , then $U(\mu)(F) \subset F$ for every $\mu \in \mathscr{C}'(G)$ . Conversely, if $U(f \cdot \beta) \subset F$ for every $f \in \mathscr{K}(G)$ , then $F$ is stable for $U$ .

The first part of (ii) is immediate, since the restrictions of the $U(s)$ to $F$ ( $s \in G$ ) define a continuous linear representation of $G$ in the quasi-complete locally convex space $F$ . The second part of (ii), and (i), follow from Cor. 3 of Lemma 4.

Exercises

See the exercises for § 2.