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J. Math. Kyoto Univ. (JMKYAZ)

46-4 (2006), 701–711

Invariant averagings of locally compact groups

By

Djavvat Khadjiev and Abdullah C¸ avus¸

Abstract

A deﬁnition of an invariant averaging for a linear representation of

a group in a locally convex space is given. Main results: A group H is

ﬁnite if and only if every linear representation of H in a lo cally convex

space has an invariant averaging. A group H is amenable if and only

if every almost periodic representation of H in a quasi-complete locally

convex space has an invariant averaging. A lo cally compact group H is

compact if and only if every strongly continuous linear representation of

H in a quasi-complete locally convex space has an invariant averaging.

Introduction

The present paper is devoted to a deﬁnition and an investigation of an in-

variant averaging for a linear representation of a group in a locally convex space.

Invariant averagings are closely connected with invariant means, vector-valued

invariant means, amenable groups, almost periodic functions, almost periodic

representations of a group in locally convex spaces and uniformly equicontinu-

ous actions of a group on compacts.

The theory of invariant means for complex-valued bounded functions on

a group was founded by von-Neumann [18], [19]. For an arbitrary complete

locally convex space L,thetheoryofL-valued almost periodic functions and L-

valued invariant means was developed by von-Neumann and Bochner [20]. The

existence of an invariant mean in the space of weakly almost periodic functions

was investigated by de-Leew and Glicksberg [14], [15]. Vector-valued invariant

means have been used by a number of authors for the study of some vector-

valued function spaces, functional equations , a linear topological classiﬁcation

of spaces of continuous functions and for solving stability problems [1], [2], [6],

[23]–[25].

It is well known[7], [4], [21]–[23] that the existence of an invariant mean on

a locally compact group G is equivalent to many fundamental properties in the

harmonic analysis of G. Below another such property will be given in terms of

invariant averagings.

2000 Mathematics Subject Classiﬁcation(s). 43A07, 43A60

This work was supported by the Research Fund of Karadeniz Technical University. Project

number:2003.11.003.1

Received January 3, 2006

702 Djavvat Khadjiev and Abdullah C¸avu¸s

Our paper is organized as follows. In section 1, a deﬁnition of an invariant

averaging for a linear representation of a group in a locally convex space is given.

It is obtained that a group H is ﬁnite if and only if every linear representation

of H in a locally convex space has an invariant averaging. In section 2, it is

proved that a group H is amenable if and only if every almost periodic linear

representation of H in a quasi-complete locally convex space has an invariant

averaging. In section 3, it is obtained that a locally compact group H is compact

if and only if every strongly continuous linear representation of H in a quasi-

complete locally convex space has an invariant averaging.

A part of results of section 1 was announced in [11].

1. The concept of an invariant averaging and invariant averagings

of ﬁnite groups

Let L be a complex locally convex space and G(L) be the group of all

continuous linear operators A : L → L such that A

−1

exists and is continuous.

Let H be a group.

Deﬁnition 1.1 ([16, p. 80]). A homomorphism α : H → G(L) will be

called a linear representation of a group H in a locally convex space L.

Let α be a linear representation of H in a locally convex space L, x ∈ L.

Put Hx = {y ∈ L : y = α(t)x, t ∈ H}. Denote the convex hull of Hx by

Conv (Hx)andtheclosureofConv (Hx)inL by V (x).

Deﬁnition 1.2. A linear operator M : L → L will be called an invariant

averaging for α if:

(i) α(t)M(x)=M(α(t)x)=M (x) for all x ∈ L and all t ∈ H;

(ii) M(x) ∈ V (x) for all x ∈ L.

Put L

H

= {y ∈ L : α(t)y = y, ∀t ∈ H}.ThenL

H

is a closed linear sub-

space of L. By Deﬁnition 1.2, M (x) ∈ L

H

for all x ∈ L, M(x)=x for all

x ∈ L

H

and M (M(x)) = M(x) for all x ∈ L. Hence M is a projection operator

onto L

H

.

Proposition 1.1. Let {K

τ

,τ ∈ T } be a family of closed α(H)-invariant

subspaces of a linear representation α of H in a locally convex space L.Assume

that α has an invariant averaging. Then

T

K

H

τ

=(

T

K

τ

)

H

,where

denotes

the algebraic sum of vector subspaces.

Proof. The inclusion

T

K

H

τ

⊂ (

T

K

τ

)

H

is evident. Prove the converse

inclusion. Let x ∈ (

T

K

τ

)

H

. Then there exist elements x

i

∈ K

τ

i

,i=1,...,m,

such that x =

m

i=1

x

i

. Applying an invariant averaging M to x, we ﬁnd

x = Mx =

m

i=1

Mx

i

.SinceMx

i

∈ K

τ

i

,i=1,...,m,wehavex ∈

T

K

H

τ

.

Invariant averagings of locally compact groups 703

Remark 1. An analog of Proposition 1.1 has an important role in the

invariant theory ([17, II.3.2], [13, II.3.2, Theorem (d)], [8], [9], [10, p. 44]).

According to Deﬁnition 1.2, V (x)containsanα(H)-invariant point for all

x ∈ L. It is very important (in particular, in the ergodic theory) to know when

V (x) has the unique α(H)-invariant point.

Deﬁnition 1.3. An invariant averaging M on L will be called continu-

ous if M is continuous on L.

Proposition 1.2. Let α be a linear representation of H in a locally

convex space L. Assume that α has a continuous invariant averaging. Then

V (x) contains the unique α(H)-invariant point for every x ∈ L.

Proof. Let M be a continuous invariant averaging for α and x ∈ L.Then

M(x) ∈ V (x)andM(x) ∈ L

H

.Lety ∈ V (x) ∩ L

H

. Then there exists a net

{y

ν

}, y

ν

∈ Conv(Hx), such that lim y

ν

= y.Everyy

ν

has the form

y

ν

= λ

(ν)

1

α(t

(ν)

1

)x + ···+ λ

(ν)

n(ν)

α(t

(ν)

n(ν)

)x,

where λ

(ν)

i

∈ R (R is the ﬁeld of real numbers), λ

(ν)

i

≥ 0and

n(ν)

i=1

λ

i

=1.

Applying the operator M to y

ν

, we ﬁnd

My

ν

= λ

(ν)

1

Mα(t

(ν)

1

)x + ···+ λ

(ν)

n(ν)

Mα(t

(ν)

n(ν)

)x = Mx

for all ν.SinceM is continuous, y = My = M (lim y

ν

) = lim My

ν

= Mx.

In the following theorem we prove that every linear representation of a

ﬁnite group in a locally convex space has a continuous invariant averaging.

Theorem 1.1. For a group H the following conditions are equivalent:

(i) H is a ﬁnite group;

(ii) every linear representation of H in a locally convex space has an in-

variant averaging.

Proof. (i) → (ii). Let H = {t

1

,...,t

n

} be a ﬁnite group, α is a linear

representation of H in a locally convex space L and x ∈ L. Consider the

operator

M(x)=

1

n

(α(t

1

)+···+ α(t

n

))(x).

It is obviously that M is an invariant averaging for α and it is continuous.

(ii) → (i). Let H be an inﬁnite group. Assume that every linear represen-

tation of H in a locally convex space has an invariant averaging.

Let Q(H) be the linear space of all complex functions on H.Denoteby

F (H) the set of all ﬁnite subsets of H. Q(H) is a locally convex space with

respect to the topology of the system {p

A

,A∈ F (H)} of semi-norms, where

p

A

(x)=max

t∈A

|x(t)|,x∈ Q(H),A∈ F (H).

704 Djavvat Khadjiev and Abdullah C¸avu¸s

Let Q

(H) be the conjugate space of Q(H). Q

(H) is a locally convex

space with respect to the w

∗

-topology. Deﬁne the linear representation α of

H in Q(H) and the linear representation α

in Q

(H) as follows: (α(h)x)(t)=

x(h

−1

t), (α

(h)ϕ)(x)=ϕ(α(h

−1

)x), where h ∈ H, x ∈ Q(H),ϕ ∈ Q

(H). Put

e

t

(s)=

1fort = s,

0fort = s

for all t, s ∈ H.Thene

t

∈ Q(H) for all t ∈ H.

Lemma 1.1.

(a) For ϕ ∈ Q

(H) the set {t ∈ H : ϕ(e

t

) =0} is ﬁnite;

(b) If {t ∈ H : ϕ(e

t

) =0} = ∅ for ϕ ∈ Q

(H) then ϕ =0.

ProofoftheLemma. (a). Onaccountofϕ ∈ Q

(H)thereexistasemi-norm

p

A

,A ∈ F(H), and c ∈ R, c > 0, such that |ϕ(x)|≤cp

A

(x). Then ϕ(e

t

)=0

for all t/∈ A. Hence {t ∈ H : ϕ(e

t

) =0}⊂A and the set {t ∈ H : ϕ(e

t

) =0} is

ﬁnite.

(b). Assume that ϕ ∈ Q

(H)andϕ(e

t

) = 0 for all t ∈ H. On account of

ϕ ∈ Q

(H)wehave|ϕ(x)|≤cp

A

(x)forsomeA ∈ F (H)andsomec ∈ R, c > 0.

Let A = {t

1

,...,t

m

}. Every element x ∈ Q(H) has the form x = x

1

+x

2

,where

x

1

(t)=x(t

1

)e

t

1

+ ···+ x(t

m

)e

t

m

,x

2

= x − x

1

. Using the inequality |ϕ(x)|≤

cp

A

(x), we ﬁnd ϕ(x

2

)=0andϕ(x

1

)=x(t

1

)ϕ(e

t

1

)+···+ x(t

m

)ϕ(e

t

m

)=0.

Hence ϕ(x) = 0 for all x ∈ Q(H). The lemma is proved.

According to supposition (ii) of our theorem there exists an invariant av-

eraging M on Q

(H). Let ϕ ∈ Q

(H) such that ϕ(1

H

)=1,where1

H

is the

function: 1

H

(t) = 1 for all t ∈ H.ThenMϕ is an α

(H)-invariant functional

on Q(H)and(Mϕ)(1

H

)=1,sinceMϕ ∈ V (ϕ). Hence Mϕ = 0. According

to Lemma 1.1 there exists s ∈ H such that (Mϕ)(e

s

) =0. SinceMϕ is α

(H)-

invariant, we have Mϕ(e

s

)=Mϕ(α(t

−1

)e

s

)=Mϕ(e

ts

) =0forallt ∈ H.But

it is a contradiction to statement (a) of Lemma 1.1. Hence H is ﬁnite. The

theorem is completed.

2. Invariant averagings of amenable groups

Let α be a linear representation of a group H in a quasi-complete locally

convex space L.

Deﬁnition 2.1. An element x ∈ L will be called almost periodic if the

orbit Hx is precompact in L. A representation α will be called almost periodic

if every element of L is almost periodic.

Remark 2. This is a variant of the deﬁnition of an almost periodic

operator semigroup proposed by K. de Leeuw and I. Glicksberg [14].

Let α be an almost periodic representation of H in a quasi-complete locally

convex space L. Then it is known [5, 8.13.4(2)] that V (x) is a compact for all

x ∈ L.

Invariant averagings of locally compact groups 705

Theorem 2.1. For a group H the following conditions are equivalent:

(i) H is an amenable group;

(ii) every almost periodic representation of H in a quasi-complete locally

convex space has an invariant averaging.

Proof. (ii) → (i). Let B(H) be the set of all bounded complex functions

on H. B(H) is a Banach space with respect to the norm:

x =sup

t∈H

|x(t)|,

where x ∈ B(H). Let B(H)

be the conjugate space of B(H). According to

Corollary 2 in [3, ch.III, 3.7] B(H)

is a quasi-complete locally convex space

with respect to the w

∗

-topology. For ϕ ∈ B(H)

put (T

s

ϕ)(x)=ϕ(x

s

), where

s ∈ H, x

s

(t)=x(s

−1

t). Then |(T

s

ϕ)(x)| = |ϕ(x

s

)|≤ϕx

s

= ϕx.

Hence T

s

ϕ≤ϕ for all s ∈ H.Wehave

n

i=1

λ

i

(T

s

i

ϕ)(x)≤

n

i=1

λ

i

(T

s

i

ϕ)(x)≤

n

i=1

λ

i

ϕx = ϕx

for λ

i

∈ R such that λ

i

≥ 0and

n

i=1

λ

i

= 1. Hence V (ϕ) is bounded in

B(H)

and it is compact with respect to the w

∗

-topology. Then according

to supposition (ii) of our theorem there exists an invariant averaging M on

B(H)

.Letµ ∈ B(H)

be a mean that is µ(x) ≥ 0 for all x ∈ B(H), x ≥ 0and

µ(1

H

)=1,where1

H

(t) = 1 for all t ∈ H.ThenM(µ)isanH-invariant mean.

Therefore H is an amenable group.

(i) → (ii). Let H be an amenable group, m is a two-sided invariant mean

of H and α is an almost periodic representation of H in a quasi-complete locally

convex space L.

Let L

be the conjugate space of L.Forx ∈ L, F ∈ L

we consider the

function ψ

x

(t)=<F,α(t)x>= F (α(t)x)onH. Since the set Hx is precompact

and L is a quasi-complete space, V (x) is compact. Hence ψ

x

∈ B(H). Put

˜m(F )=m(ψ

x

)=m(<F,α(t)x>). Then ˜m is a linear functional on L

.We

write ˜m in the form ˜m(F )=<M(x),F >, M(x) ∈ (L

)

∗

,where(L

)

∗

is the

algebraic conjugate space of L

. The mapping M : L → (L

)

∗

is linear. Prove

that M(x) ∈ L for all x ∈ L.

Let Σ = {µ ∈ B(H)

: µ(1

H

)=1,µ(x) ≥ 0, ∀x ≥ 0} be the set of all means

on B(H). For f ∈ B(H)andt

i

∈ H put δ

t

i

(f)=f(t

i

). Let Σ

0

be the set of all

µ ∈ Σ such that: µ =

n

i=1

λ

i

δ

t

i

,λ

i

≥ 0,

n

i=1

λ

i

=1forsomet

i

∈ H, i =1,...,n.

For µ ∈ Σ

0

we consider the operator M

µ

(x)=

n

i=1

λ

i

α(t

i

)x,whereµ =

n

i=1

λ

i

δ

t

i

.

There exists a net {µ

ν

} ,µ

ν

∈ Σ

0

, such that lim µ

ν

= m in the w

∗

-topology

in B(H)

.WehaveM

µ

ν

(x) ∈ V (x) for all ν.Putx

ν

= M

µ

ν

(x). On account

of compactness of V (x) there exist a subnet {y

τ

} of {x

ν

} and x

0

∈ V (x)such

that lim y

τ

= x

0

. Then lim µ

τ

= m, µ

τ

(<F,α(t)x>)=<M

µ

τ

(x),F >,

706 Djavvat Khadjiev and Abdullah C¸avu¸s

µ

τ

(<F,α(t)x>) → m(<F,α(t)x>) for all F ∈ L

.Using<M

µ

τ

(x),F >→

<x

0

,F >,weobtainm(<F,α(t)x>)=<M(x),F >=<x

0

,F > for all

F ∈ L

.ThenM(x)=x

0

∈ L.ThusM(x) ∈ L and M(x) ∈ V (x).

Prove α(s)M (x)=M(α(s)x)=M (x). Since m is H-invariant, we ﬁnd

<F,α(s)M (x) >= m(<F,α(s)α(t)x>)=m(<F,α(t)x>)=<F,M(x) >.

Similarly <F,M(α(s)x) >= m(<F,α(t)α(s)x>)=m(<F,α(t · s)x>)=

m(<F,α(t)x>)=<F,M(x) >.

3. Invariant averagings of locally compact groups

Deﬁnition 3.1. A linear representation α : H → G(L) of a topological

group H in a locally convex space will be called strongly continuous if t → α(t)x

is a continuous function on H for every x ∈ L.

Our aim in this section is a proof of the following

Theorem 3.1. For a locally compact group H the following conditions

are equivalent:

(i) H is compact;

(ii) every strongly continuous linear representation of H in a quasi-complete

locally convex space has an invariant averaging.

Proof. The implication (i) → (ii) is known [12, p. 149].

A proof of the implication (ii) → (i) consists of some steps. First we give

some needful lemmas.

Let H be a topological group, α is a strongly continuous linear represen-

tation of H in a complex locally convex space L and L

is the conjugate space

of L. We deﬁne a linear representation of H in L

as follows:(α

(t)ϕ)(x)=

ϕ(α(t

−1

)x).

Lemma 3.1. α

is a strongly continuous linear representation with re-

spect to the topology σ(L

,L).

ProofoftheLemma. ItisknownthatL

is a locally convex space with

respect to the topology σ(L

,L). For x

1

,...,x

n

∈ L and ε ∈ R, ε > 0, put

Q(x

1

,...,x

n

,ε)={ϕ ∈ L

: |ϕ(x

i

)| <ε, i=1,...,n}. The family

{Q(x

1

,...,x

n

,ε),x

i

∈ L, ε ∈ R, ε > 0,n∈ N}

is a fundamental system of neighborhoods of the zero in L

for the topol-

ogy σ(L

,L). From α

(t)Q(x

1

,...,x

n

,ε)=Q(α(t

−1

)x

1

,...,α(t

−1

)x

n

,ε)and

α

(t

−1

)Q(x

1

,...,x

n

,ε)=Q(α(t)x

1

,...,α(t)x

n

,ε), we obtain that operators

α

(t)andα

(t

−1

) are continuous in the the topology σ(L

,L) for every t ∈ H.

Let ϕ be a ﬁxed element of L

and Q(x

1

,...,x

n

,ε) be an arbitrary neigh-

borhood of the zero in L

. For arbitrary ε>0thereexistsaneighborhood

W of the zero in L such that |ϕ(W )| <ε.Sinceα is strongly continuous, for

x

1

,...,x

n

∈ L, the neighborhood W of the zero in L and every t

0

∈ H there

exists a neighborhood U of the unit in H such that α(Ut

0

)x

i

−α(t

0

)x

i

⊂ W for

Invariant averagings of locally compact groups 707

all i =1,...,n.Then|ϕ(α(Ut

0

)x

i

− α(t

0

)x

i

)| <εfor all i =1,...,n. Hence

|α

(U

−1

t

−1

0

)ϕ(x

i

) − α

(t

−1

0

)ϕ(x

i

)| <εfor all i =1,...,n.Thenα

(U

−1

t

−1

0

)ϕ −

α

(t

−1

0

)ϕ ⊂ Q(x

1

,...,x

n

,ε). This means that the mapping t → α

(t)ϕ is con-

tinuous for every t

0

∈ H and every ϕ. Thus the representation α

is strongly

continuous. The lemma is proved.

Let H be a locally compact group. Denote by K(H) the vector space of

all complex continuous functions on H with the compact support. Denote the

family of compact subsets of H by T (H). For A ∈ T (H)andx ∈ K(H) put

p

A

(x)=max

t∈A

|x(t)|. According to [5, Theorem 6.31] K(H) is a barrel locally

convex space with respect to the family {p

A

,A∈ T (H)}.

Denote the vector space of all complex continuous functions on H by C(H).

C(H) is a locally convex space with respect to the family {p

A

,A∈ T (H)} of

semi-norms. We have K(H) ⊂ C(H).

Lemma 3.2. K(H) is dense in C(H).

Proof of the Lemma. According to statement 0.2.18(2)in [5] for every

A ∈ T (H) there exist a function e

A

∈ K(H) and a compact neighborhood U

of A such that

e

A

(t)=

1fort ∈ A,

0fort/∈ U.

Let x ∈ C(H). Then e

A

x ∈ K(H)andp

A

(x − e

A

x) = 0. Therefore for the

net {e

A

x, A ∈ T (H)} we obtain lim

A∈T (H)

e

A

x = x. The lemma is proved.

Using Lemma 3.2 and [5, Theorem 6.2.4(2)], we obtain the following

Lemma 3.3. C(H) is a barrel locally convex space.

We deﬁne a linear representation of H in C(H) as follows: (α(s)x)(t)=

x(s

−1

t),x∈ C(H).

Lemma 3.4.

(i) The linear representation α in C(H) is strongly continuous;

(ii) The conjugate space (C(H))

is a quasi-complete locally convex space

with respect to the topology σ(L

,L),whereL = C(H);

(iii) The linear representation α

in (C(H))

is strongly continuous.

Proof of the Lemma. Put W

A,ε

= {x ∈ C(H):p

A

(x) <ε}.Fors ∈ H

we have α(s)W

A,ε

= W

sA,ε

. Hence operator α(t) is continuous on L for every

t ∈ H.Weprovethatα is strongly continuous. Let x ∈ C(H), s

0

∈ H and A ∈

T (H). On account of compactness of A there exists a neighborhood U of the

unit of H such that |x(Us

−1

0

t)− x(s

−1

0

t)| <εfor all t ∈ A.Thenp

A

(α(Us

0

)x−

α(s

0

)x) <ε. This means that the mapping t → α(t)x is continuous on H for

every x.Thusα is strongly continuous.

Using Lemma 3.3 and [3, III.3.7, Corollary 2], we ﬁnd that the conjugate

space (C(H))

is a quasi-complete locally convex space with respect to the

σ(L

,L)- topology, where L = C(H). According to Lemma 3.1 the linear

representation α

in (C(H))

is strongly continuous with respect to the σ(L

,L)-

topology. The lemma is proved.

For x ∈ C(H) put supp(x)={t ∈ H : x(t) =0}.

708 Djavvat Khadjiev and Abdullah C¸avu¸s

Lemma 3.5. Let H be a locally compact topological group, A ∈ T (H)

and U be an arbitrary closed neighborhood of the unit of H. Then there exist a

family {e

k

(t) ∈ C(H),k=0, 1,...,n} and elements t

1

,...,t

n

∈ H such that:

(i)

n

k=0

e

k

(t)=1and 0 ≤ e

k

(t) ≤ 1 for all t ∈ H, k =0, 1,...,n;

(ii) supp(e

0

) ∩ A = ∅;

(iii) A ⊂∪

n

k=1

t

k

U and supp(e

k

) ⊂ t

k

U for all k =1,...,n.

Proof of the Lemma. We consider the following open covering of the

compact set A: A ⊂∪

t∈A

tU. Then there exist t

1

,...,t

n

∈ A such that

A ⊂∪

n

k=1

t

k

U.PutB = H \∪

n

k=1

t

k

U. H is a completely regular topologi-

cal space as a separable topological group. Hence for t

k

and the neighborhood

t

k

U there exists a continuous real function f

k

: H → R such that 0 ≤ f

k

(t) ≤ 1

for all t ∈ H, f

k

(t

k

)=1andf

k

(t) = 0 for all t ∈ H \t

k

U.Putf

k

(t)=1−f

k

(t).

We consider the multiplication

(f

1

(t)+f

1

(t))(f

2

(t)+f

2

(t)) ···(f

n

(t)+f

n

(t)) = 1.

By induction we obtain

f

1

(t)+f

1

(t)f

2

(t)+f

1

(t)f

2

(t)f

3

(t)+···+ f

1

(t)f

2

(t) ···f

n−1

(t)f

n

(t)

+ f

1

(t)f

2

(t) ···f

n

(t)=1.

Put e

1

= f

1

(t),e

i

= f

1

(t)f

2

(t) ···f

i−1

(t)f

i

(t),i=2,...,n; e

0

=

f

1

(t)f

2

(t) ···f

n

(t). Then supp(e

0

) ⊂ H \∪

n

k=1

t

k

U and supp(e

k

) ⊂ t

k

U for

all k =1,...n. The lemma is proved.

Lemma 3.6. Let H be a locally compact topological group such that

there exists a non-zero H-invariant linear continuous functional ϕ on C(H).

Then H is compact.

Proof of the Lemma. Assume that H is non- compact and ϕ be a non-zero

H-invariant continuous linear functional on C(H). Since ϕ is continuous, there

exist c ∈ R, c > 0, and A ∈ T (H) such that |ϕ(f)|≤cp

A

(f) for all f ∈ C(H).

Then ϕ(f)=0forf ∈ L such that supp(f) ∩ A = ∅.SinceH is non-compact,

there exist y ∈ H and a closed neighborhood U of the unit of H such that

A ∩ yU = ∅.

Since ϕ is non-zero, ϕ(f) =0forsomef ∈ C(H). AccordingtoLemma

3.5 there exist elements t

k

∈ H and e

k

∈ C(H),k =1,...,n, satisfying the

conditions (i), (ii), (iii) of Lemma 3.5. Then ϕ(e

j

f) =0andsupp(e

j

(t)f(t)) ⊂

t

j

U for some j.Putz = yt

−1

j

.Wehavesupp(α(z))(e

j

(t)f(t)) ⊂ zt

j

U = yU.

Then supp(α(z)(e

j

(t)f(t))) ∩ A = ∅. Hence ϕ(α(z)(e

j

f)) = 0. Since ϕ is H-

invariant, ϕ(e

j

f)=ϕ(α(z)(e

j

f)) = 0. It is a contradiction. The lemma is

proved.

We continue a proof of the implication (ii) → (i) of our theorem. Consider

a linear representation α

of H in (C(H))

. According to Lemma 3.4 (C(H))

is a quasi-complete locally convex space and α

is a strongly continuous linear

Invariant averagings of locally compact groups 709

representation of H in (C(H))

. On account of supposition (ii)ofourtheorem

α

has an invariant averaging. Let a ∈ H.Putϕ

a

(x)=x(a) for all x ∈ C(H).

Then ϕ

a

∈ (C(H))

and ϕ

a

(1

H

)=1,where1

H

is the unit function of C(H).

For every s ∈ H we ﬁnd α

(s)ϕ

a

= ϕ

sa

and ϕ

sa

(1

H

)=1. Wehaveϕ(1

H

)=1

foreverylinearfunctionalϕ on C(H)oftheformϕ =

n

k=1

λ

k

α

(s

k

)ϕ

a

,where

λ

k

≥ 0,

n

k=1

λ

k

= 1. Hence ϕ(1

H

) = 1 for every ϕ ∈ V (ϕ

a

). Let M be an

invariant averaging for α

. Then the element ψ = M(ϕ)isH-invariant and

ψ(1

H

) = 1. Therefore ψ is a non-zero H-invariant linear functional on C(H).

By Lemma 3.6, H is compact. The theorem is completed.

We note that a left H-invariant Haar’s integral on a locally compact group

H is a H-invariant linear functional on K(H). Hence Lemma 3.6 has also the

following

Corollary 3.1. Haar’s integral is continuous on K(H) with respect to

the topology {p

A

,A∈ T (H)} if and only if H is compact.

Department of Mathematics

Karadeniz Technical University

61080, Trabzon, Turkey

e-mail: djavvat@yahoo.com

cavus@ktu.edu.tr

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