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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 locally 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 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.
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
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
(t
(ν)
1
)x + ···+ λ
(ν)
n(ν)
(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
tA
|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) 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.Then is an α
(H)-invariant functional
on Q(H)and()(1
H
)=1,since V (ϕ). Hence = 0. According
to Lemma 1.1 there exists s H such that ()(e
s
) =0. Since is α
(H)-
invariant, we have (e
s
)=(α(t
1
)e
s
)=(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
tH
|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
)|≤ϕx
s
= ϕx.
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
ϕx = ϕx
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(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
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
}
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
)={ϕ L
: |ϕ(x
i
)| , i=1,...,n}. The family
{Q(x
1
,...,x
n
),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
)=Q(α(t
1
)x
1
,...,α(t
1
)x
n
)and
α
(t
1
)Q(x
1
,...,x
n
)=Q(α(t)x
1
,...,α(t)x
n
), 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
) be an arbitrary neigh-
borhood of the zero in L
. For arbitrary ε>0thereexistsaneighborhood
W of the zero in L such that |ϕ(W )| .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
)| for all i =1,...,n. Hence
|α
(U
1
t
1
0
)ϕ(x
i
) α
(t
1
0
)ϕ(x
i
)| for all i =1,...,n.Thenα
(U
1
t
1
0
)ϕ
α
(t
1
0
)ϕ Q(x
1
,...,x
n
). 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
tA
|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
AT (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) }.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)| for all t A.Thenp
A
(α(Us
0
)x
α(s
0
)x) . 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 ⊂∪
tA
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)=1f
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
n1
(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
i1
(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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Article
Main results: For every equicontinuous almost periodic linear representation of a group in a complete locally convex space L with the countability property, there exists the unique invariant averaging; it is continuous and is expressed by using the L-valued invariant mean of Bochner and von-Neumann. An analog of Wiener's approximation theorem for an equicontinuous almost periodic linear representation in a locally convex space with the countability property is proved.
Article
We show that a Banach space X is complemented in its ultraproducts if and only if for every amenable semigroup S the space of bounded X-valued functions defined on S admits (a) an invariant average; or (b) what we shall call “an admissible assignment”. Condition (b) still provides an equivalence for quasi-Banach spaces, while condition (a) necessarily implies that the space is locally convex.