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Journal of Sciences, Islamic Republic of Iran 18(1): 41-48 (2007) University of Tehran, ISSN 1016-1104

The Topological Center of the Banach Algebra UCl(K)* R.A. Kamyabi-Gol* Department of Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Islamic Republic of Iran

Abstract Let K be a (commutative) locally compact hypergroup with a left Haar measure. Let L1(K) be the hypergroup algebra of K and UCl(K) be the Banach space of bounded left uniformly continuous complex-valued functions on K. In this paper we show, among other things, that the topological (algebraic) center of the Banach algebra UCl(K)* is M(K), the measure algebra of K. Keywords: Hypergroup; Hypergroup algebra; Measure algebra; Second conjugate algebra; Algebraic center

equal almost everywhere identified), and the multiplication defined by

1. Introduction The theory of hypergroups was initiated by Dunkl [4], Jewett [8] and Spector [21] in the early 1970's and has received a good deal of attention from harmonic analysts (note that Jewett calls hypergroups ''convos'' in his paper [8]). In [16], Pym also considers convolution structures which are close to hypergroups. A fairly complete history is given in Ross's survey article [17,18]. Hypergroups arise in a natural way as a double coset space, and the space of conjugacy classes of a compact group [17,1]. In particular, locally compact groups are hypergroups. Here we follow the method of Jewett [8]. It is still unknown if an arbitrary hypergroup admits a left Haar measure but all the known examples do [8, §5]. In particular, discrete, compact and commutative hypergroups possess Haar measures [10]. Throughout, K will denote a hypergroup with a left Haar measure λ . Let L1 (K ) denote the hypergroup algebra of K , i.e. all Borel measurable functions φ on K

with

φ1 =

∫

φ *ψ (x ) =

∫

φ (x * y )ψ ( y )d λ ( y )

(see [8, §5.5]).

K

Let the second dual L1 (K )** ( = L∞ (K )* ) of L1 (K ) be equipped with the first Arens product [3]. Then L1 (K )** is a Banach algebra with this product. The topological center of L1 (K )** is defined by Z (L1 (K )** ) = {m ∈ L1 (K )** : the mapping n

mn

is w*-continuous on L1 (K )** }. We have shown [9] that the topological center of L1 (K )** is L1 (K ) . This fact has been shown by Lau and Losert in [13] for locally compact groups (see also [14] and [2]). Let UC l (K ) be the Banach space of all bounded left uniformly continuous complex-valued functions on K (see Section 2 for definition) and UC l (K )* be its dual Banach space. Then there is a natural multiplication on UC l (K )* under which it is a Banach algebra. More

| φ (x ) |d λ (x ) < ∞ (with functions

K

2000 Mathematics Subject Classification. Primary 43A20, 43A10, 43A22 Secondary 46H05, 46H10 * E-mail: [email protected]

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i ∈ I } are pairwise disjoint. (b) C i * y i * y j ∩ C p * z p * z q = ∅ ,

specifically, for m , n ∈UC l (K )* , f ∈UC l (K ) , and x ∈K ,

This product is, in fact, the restriction of the first Arens product on L1 (K )** to UC l (K )* , which will be proved in Lemma 3.1. The topological center of UC l (K )* is defined by

n when

the

xf

mapping

K

mn is w -continuous on UC l (K ) }. Note that is commutative, then

K

Z (UC l (K )* )

is

f x (y ) = f (y * x ) =

*

precisely the algebraic center of UC l (K ) . For a locally compact group G , Lau in [12] has shown that

xf

f (t ) d (δ x * δ y )(t ),

∫

K

f (t ) d (δ y * δ x )(t ),

if the integrals exist. We write

*

Z (UC l (G ) ) is M (G ) , the algebra of bounded regular Borel measures on G . However the method of his proof cannot be applied to hypergroups in general. The purpose of this paper is to establish these results for hypergroups. Our proof also provides a new proof of Lau's result [12, Theorem 1] in the group case. This paper is organized as follows: Section 2 consists of some notations and preliminary results that we need in the sequel. The technical Lemma 2.7 in this section plays a key role in proving our main result (Theorem 3.11). In Section 3, we shall prove that the topological center of UC l (K )* is M (K ) . The results of this section generalize the corresponding ones for locally compact groups [12].

y( xf

x *y

and f x * y for

f

) and (f y ) x , respectively.

The function f

is given by f (x ) = f (x ) . The

∫

∫

integral …d λ (x ) is often denoted by …dx . p

Let (L (K ), .

p)

, 1 ≤ p ≤ ∞ , denote the usual L p

spaces on K [8, §6.2]. Then L∞ (K ) is a commutative Banach algebra with pointwise multiplication and the essential supremum norm . ∞ , and moreover, L∞ (K ) = L1 (K )* [8, §6.2]. We say that X ⊆ L∞ (K ) is

translation invariant if x f ∈ X and f x ∈ X for all f ∈ X , x ∈ K ; also X is topologically translation invariant if φ * f ∈ X and f * φ ∈ X

for all f ∈ X ,

φ ∈ P (K ) = {φ ∈ L (K ) : φ ≥ 0 , φ 1= 1} . In addition, we use the following notations: C 00 (K ) : the set of continuous functions with compact supports on K . C (K ) : the set of bounded continuous functions on K . UC l (K ) = {f ∈ C (K ) : x x f is continuous from K into (C (K ), . ∞ )} . UC r (K ) = {f ∈ C (K ) : x f x is continuous from K into (C (K ), . ∞ )} . It is known that UC l (K ) = { f ∈ C (K ) : x x f is continuous from K into C(K) with the weak-topology} [20, Theorem 4.2.2, p. 88]. Each of the spaces UC l (K ) and UC r (K ) is a normed closed, conjugate closed, translation invariant and topologically translation invariant subspace of C (K ) containing the constant functions and C 0 ( K ) 1

2. Preliminaries and Some Technical Lemmas The notations used in this paper are those of [8] with the following exceptions: The mapping x → x denotes the involution on the hypergroup K , δ x the Dirac measure concentrated at x ( x ∈ K ), and 1X the characteristic function of the non-empty set X ⊆ K . For C ⊆ K and y ∈ K , C * y denotes the subset C *{ y } of K . Lemma 2.1. Let K be a locally compact non-compact hypergroup. Then there exists a family {C i : i ∈ I } of compact subsets of K, and y i , z i ∈ K , for each i ∈ I such that C i

∫

( y ) = f (x * y ) =

and

and f x is the right translation

*

*

i ≠ j

p ≠ q , i , j , p ,q ∈ I . Proof. See [9, Lemma 2.1]. □ For a Borel function f on K and x ∈ K , denotes the left translation

〈 mn , f 〉 = 〈 m , nf 〉 where nf (x ) = 〈 n , x f 〉 .

Z (UC l (K )* ) = {m ∈UC l (K )* :

J. Sci. I. R. Iran

(the interior of C i ) is non-empty,

∪i ∈I C i = K , {C i : i ∈ I } is closed under finite unions, and (a) the families {C i * y i : i ∈ I } and {C i * z i :

42

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Vol. 18 No. 1 Winter 2007

m ∈ A ** , the following are

[19, Lemma 2.2]. Furthermore (i) UC l (K ) = L1 (K ) *UC l (K ) = L1 (K ) * L∞ (K ) [19, Lemma 2.2] (ii) UC r (K ) = UC l (K ) * L1 (K ) = L∞ (K ) * L1 (K ) [19, Lemma 2.2]. Note that UC l (K ) is not an algebra in general [19, Remark 2.3(b)]. For φ ∈ L1 (K ), we write φ (x ) = Δ (x )φ (x ) where

Lemma 2.3 For any equivalent: (a) m ∈ Z (A ** ) ;

Δ is the modular function on K ; then φ = φ

Proof. [3, p. 313]. □ Note that for n fixed in A ** , the mapping m mn is always w * - w * continuous. We collect here some facts about the Arens product on L1 (K )** that we shall need later.

f ∈ L p (K ) , 1 ≤ p ≤ ∞ , x ∈ K , then

xf

p

1.

(b) the map n → mn from A ** into A ** is w * - w * continuous; (c) the map n → mn from A ** into A ** is w * - w * continuous on norm bounded subsets of A ** .

If

≤ f ,

and this is in general not an isometry [8, §3.3]. The is continuous from K to mapping x xf (L p (K ), .

p)

, 1 ≤ p < ∞ , [8, 2.2B and 5.4H]. Lemma 2.4. Let φ ,ψ ∈ L1 (K ) , f ∈ L∞ (K ) . Then (i) 〈ψ f , φ 〉 = 〈 f φ ,ψ 〉 .

It is easy to show that L1 (K ) has a bounded + (K ) such approximate identity (B.A.I) {e i : i ∈ I } ⊆ C 00 that e i = 1 (see [19, Lemma 2.1]). For any Banach space X , we denote its first and second dual by X * and X ** . Let A be a Banach algebra. For any f ∈ A * and a ∈ A , we may define a linear functional fa on A by 〈 fa, b 〉 = 〈 f , ab 〉 , (b ∈ A ).

One can check that fa ∈ A * and

fa ≤ f

(ii) ψ f = f *ψ ∈UC r (K ) , f φ = φ * f ∈UC l (K ) . (iii) a (ψ f ) = ψ ( a f ) , (f φ )a = (f a )φ for a ∈ K . Proof. immediate. □ Lemma 2.5. Let 0 ≠ m ∈ L∞ (K )* . Then there is a net

a . Now

{u α } in L1 (K ) such that

for n ∈ A ** , we may define nf ∈ A * by 〈 nf , a 〉 = 〈 n , fa〉 ; clearly we have nf ≤ n f . Next for

compact support and u α → m in the w * -topology of L∞ (K )* .

m ∈ A ** , define mn ∈ A ** by 〈 mn , f 〉 = 〈 m , nf 〉. We

have mn ≤ m n , and A ** becomes a Banach algebra with the multiplication mn , just defined, referred to as the first Arens product. There is another multiplication on A ** , called the second Arens product, which is denoted by m n and defined successively as follows: 〈 m n , f 〉 = 〈 n , fm 〉 , where 〈 fm , a 〉 = 〈 m , af 〉 , 〈af , b 〉 = 〈 f , ba 〉 , and m , n , f , a , b are taken as above.

Proof. This follows from Goldstine's theorem and the density of C 00 (K ) in L1 (K ). □ Lemma 2.6. If m ∈ Z (L1 (K )** ) and f ∈ L∞ (K ) , then fm ∈UC l (K ) and (fm )(x * y ) = 〈 m , f x * y 〉 . Proof. See [9, Lemma 2.6]. □ Lemma 2.7. If n ∈ Z (L1 (K )** ) and u ∈ L1 (K ) are such that (n − u )(f ) = 0 for all f ∈ C 0 (K ) , then n =u .

From now on A ** will always be regarded as a Banach algebra with the first Arens product. Let Z (A ** ) denote the set of all m ∈ A ** such that mn = m n for all n ∈ A ** . We call Z (A ** ) the

Proof. See [9, Lemma 2.7]. □

topological center of A ** . **

Lemma 2.2. Z (A ) is a closed subalgebra of A containing A .

u α ≤ m , all u α have

3. Topological Center of UCl(K)*

**

In this section we show that the topological center of UC l (K )* is M (K ) . Let f ∈UC l (K ) and m ∈ UC l (K )* . Define the function mf

Proof. [3, p. 310] or [13, Lemma 1]. □

43

on K by mf (x )

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Kamyabi-Gol

= 〈 m , x f 〉 . Then mf ∈UC l (K ) . Indeed, it is easy to

Note that we can even identify UC l (K )* as a closed

see that mf ∈ C (K ) . Also

right ideal of the Banach algebra L∞ (K )* with the first Arens product (see [14, p. 13]).

(mf )( y ) = mf (x * y )

x

=

∫

mf (t )d (δ x * δ y )(t )

∫

〈 m , t f 〉d (δ x * δ y )(t )

x *y

=

x *y

= 〈m ,

∫

x *y

∫

x *y

∫

x *y

tf

Lemma 3.2. If we take C 0 (K ) ⊥ = {m ∈UC l (K )* :

d (δ x * δ y )(t )〉

tf

But the Bochner integral y( xf

J. Sci. I. R. Iran

tf

m |C (K ) = 0} , then UC l (K )* = C 0 (K ) ⊥ ⊕ M (K ) . If 0 m ∈UC l (K )* and m = m1 + μ for m1 ∈ C 0 (K ) ⊥ and

μ ∈ M (K ) , then m = m1 + μ and C 0 (K ) ⊥ is a

(*)

closed ideal in UC l (K )* .

d (δ x * δ y )(t ) is

Proof. See [15, Theorem 4]. □

) since d (δ x * δ y )(t )(ξ ) = 〈δξ , =

∫

x *y

=

∫

x *y

=

∫

x *y

∫

x *y

tf

Remark 3.3. For m ∈UC l (K )* and f ∈UC l (K ), we may define a bounded complex function fm on K by fm (x ) = 〈 m , f x 〉 . Generally, fm is not in UC l (K ) but for m = δ a ( a ∈ K ) fm = f δ a = a f ∈UC l (K ) . If

d (δ x * δ y )(t )〉

〈δξ , t f 〉d (δ x * δ y )(t )

n ∈UC l (K )* and fm ∈UC l (K ) , for all f ∈UC l (K ) , tf

(ξ )d (δ x * δ y )(t )

then we may define another product on UC l (K )* by 〈 m n , f 〉 = 〈 n , fm 〉 .

f ξ (t )d (δ x * δ y )(t )

m ∈UC l (K )

*

= f ξ (x * y ) = y ( x f )(ξ ).

)( y ) = 〈 m , y ( x f )〉 = m ( x f )( y ),

m , n ∈UC l (K )*

for

all

norm bounded subset of UC l (K )* }. Lemma 3.5. M (K ) ⊆ C .

Now we may define a product on UC l (K )* by for

fm ∈UC l (K )

Put C = {m ∈UC l (K )* : L m is w*-w* continuous on

.

Note that if m = δ a for some a ∈ K , then δ a f = f a . 〈 nm , f 〉 = 〈 n , mf 〉

all

L m (n ) = mn , n ∈UC l (K )* .

x ( mf ) − y ( mf ) ≤ m ( x f ) − m ( y f )

−yf

of

L m from UC l (K )* into itself by

Hence

xf

set

Note 3.4. For m ∈UC l (K )* , define the linear operator

(1)

) = m ( x f ).

≤ m

that

the

One can check that Z (UC l (K )* ) contains all point evaluation functionals δ x , x ∈ K .

that is, x ( mf

such

denote

f ∈UC l (K ) and mn = m n for all n ∈UC l (K )* .

So (*) implies that x ( mf

Z (UC l (K )* )

Let

and

f ∈

Proof. For μ ∈ M (K ) , we need to show that the map

UC l (K ) . With this product, one can see that UC l (K )* is a Banach algebra. Lemma 3.1 The product on UC l (K )* is the restriction of the first Arens product on

m → μ m is w * - w * continuous on any norm bounded

subset of UC l (K )* . Let {mα } be a net in UC l (K )*

L∞ (K )* to UC l (K )* .

with

Proof. See [15, Theorem 7]. □

for

mα ≤ c , for some constant c , converging to

m ∈UC l (K )* in the w * -topology of UC l (K )* . Then

44

any

f ∈UC l (K )

and

s ,t ∈ K ,

we

have

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Kamyabi-Gol

| mα f (s ) − mα f (t ) | = | 〈 mα , s f −t f 〉 |≤ c

s

f − tf .

2.4(iii), for every φ ∈ L1 (K ) ⊆ C (Lemma 3.5), (*) implies that

Hence by [11, p. 232] the family {mα f } in UC l (K ) is equicontinuous. Since mα f → mf pointwise on K , the convergence is uniform on every compact set in K [11, Theorem 7.15]. Let μ ∈ M (K ) be with compact support, then 〈 μ mα − μ m , f 〉 = 〈 μ , mα f − mf 〉 =

∫

K

Vol. 18 No. 1 Winter 2007

〈φ ,

∫

x *y

f t d δ x * δ y (t )〉 = f φ (x * y ) = (f φ ) y (x ) = ((f y )φ )(x ) = ((f y )φ ) x (e ) = (f y ) x φ (e ) = 〈φ , (f y ) x )〉.

(mα f − mf )(x )d μ (x ) → 0 . Since measures with

Hence from (*) we have 〈 m , f x * y )〉 = fm (x * y ). □

compact supports are norm dense in M (K ) and mα f ≤ c f , it follows that μ mα → μ m in the w * -

Lemma 3.7. For each m ∈UC l (K )* the following are equivalent: (a) m ∈ Z (UC l (K )* ) ,

topology of UC l (K )* and we are done. □ Lemma 3.6. If m ∈ C and f ∈UC l (K ), then fm ∈ C (K ) and fm (x * y ) = 〈 m , f x * y 〉 for all

(b) The operator L m is w * - w * continuous, (c) m ∈ C .

x ,y ∈K.

Proof. First we show that (a) implies (b). Let {nα } be

Proof. If {x α } is a net in K converging to x , then

a net in UC l (K )* converging to n ∈UC l (K )* in the

the net {δ x } converges to δ x in the w * -topology of α

*

UC l (K ) Hence

w * -topology of f ∈UC l (K ) ,

(see [8, Lemma 2.2B] and Lemma 3.2).

Then

for

every

limα mnα (f ) = limα 〈 mnα , f 〉

fm (x α ) = 〈 m , f x 〉 = 〈 m , δ x f 〉 α

UC l (K )* .

α

= limα 〈 m nα , f 〉 = limα 〈 nα , fm 〉

= 〈m δ x , f 〉 → 〈m δ x , f 〉

= 〈 m n , f 〉 = mn (f ).

α

= 〈 m , δ x f 〉 = 〈 m , f x 〉 = fm (x ),

since m ∈ C and {δ x } is bounded. Furthermore, we

(b) clearly implies (c). To show that (c) implies (a), let m ∈ C and f ∈UC l (K ) , then by Lemma 3.6, fm ∈ C (K ) . To see

know that fm is also bounded. Consequently fm ∈ C (K ). Note that for every x , y ∈ K , the

that fm ∈UC l (K ) , we first show that if θ ∈ C (K )* and a ∈ K , then

α

Bochner's integral

∫

x *y

f t d (δ x * δ y ) exists. Indeed, the

〈θ , a (fm )〉 = 〈 m δ aθ , f 〉.

map t → f t from the compact subset x * y of K into UC l (K ) is continuous in the topology σ (UC l (K ),C ) of UC l (K ), and C separates the points of UC l (K ) ( C contains the point evaluations). Hence for any m ∈C 〈m ,

∫

x *y

f t d (δ x * δ y )(t )〉 =

∫

〈 m , f t 〉d (δ x * δ y )(t )

=

∫

fm (t )d (δ x * δ y )(t ) (*)

x *y

x *y

(**)

Indeed, for θ = δ x ( x ∈ K ), by Lemma 3.6, we have 〈δ x , a (fm )〉 = a (fm )(x ) = fm (a * x ) = 〈 m , f a*x 〉 = 〈 m , (f x )a 〉 = 〈 m , δ a (f x )〉 = 〈 m δ a , f x 〉 = 〈 m δ a , δ x f 〉 = 〈 m δ a δ x , f 〉.

If θ

= fm (x * y ).

is a mean on C (K ) , then there is

nβ

integral

θ β = Σi =1λi δ x , a convex combinations of point

f t d (δ x * δ y )(t ) is equal to f x * y . By using Lemma

evaluations, such that θ β → θ in the w * -topology of

On

∫

x *y

the

other

hand,

the

Bochner's

i

45

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Kamyabi-Gol

Proof. It is easy to check that π is w * - w * continuous. For the second part, we first define a continuous map f Ff of L∞ (K ) into itself for each

C (K )* . Hence 〈θ , a (fm )〉 = lim β 〈θ β , a (fm )〉

F ∈UC l (K )* . Note that for f ∈ L∞ (K ), φ ∈ L1 (K ), we

= lim β 〈 m δ aθ β , f 〉 = 〈 m δ aθ , f 〉. *

by w - w

*

know that f φ ∈UC l (K ) (Lemma 2.4(ii)), so φ 〈 F , f φ 〉 is a continuous linear functional on

continuity of L m on norm bounded subsets

L1 (K ) and therefore corresponds to an element Ff of

*

of UC l (K ) . Consequently (**) holds. Now to see that fm ∈UC l (K ) , by [20, Theorem 4.2.2, p. 88], it is enough to show that the map x →x (fm ) from K to C (K ) is weakly continuous. Let {x α } be a net in K converging to x and

L∞ (K ) . The adjoint of φ

α

x α (fm )〉

w -continuous map m

Thus for φ ∈ L1 (K ) , f ∈ L∞ (K ) , F ∈UC l (K )* , and m ∈ L∞ (K )* ,

〈 Ff , φ 〉 = 〈 F , f φ 〉 , 〈 mF , f 〉 = 〈 m , Ff 〉 (*).

= lim〈 m x θ , f 〉 α

α

Let {φi } ⊆ L1 (K ) be a net converging to m in the

= 〈 m δ x θ , f 〉 = 〈θ , x (fm )〉,

w * -topology of L∞ (K )* then for each f ∈ L∞ (K ), by (*),

by w * - w * continuity of L m on norm bounded subsets

〈 mn , f 〉 = lim〈φi n , f 〉 = lim〈φi n , f 〉

of UC l (K )* . Hence, fm ∈UC l (K ). If n is a mean on UC l (K ) , there exists a net l

nα = Σiα=1λi δ x

i

i

of UC l (K )* . Hence for f ∈UC l (K ), considering Remark 3.3, we have

i

Lemma 3.10. Z (UC l (K )* ) = { m ∈UC l (K )* : φ m ∈ Z (L∞ (K )* ) for each φ ∈ L1 (K ) }.

= limα 〈 nα , fm 〉 = limα 〈 m nα , f 〉

Proof. Let φ ∈ L1 (K )

= limα 〈 mnα , f 〉 = 〈 mn , f 〉

n ∈UC l (K )* ,

all

enough to show that n → φ mn is w * - w * continuous.

m∈

i.e.

If nα → n in the w * -topology of L∞ (K )* , then

π ( nα )

Remark 3.8. For φ ∈ L (K ) and m ∈UC l (K ) , the product φ m makes sense both as an element of *

∞

*

π (n ) (since π is w * - w * continuous) in the

w * -topology of UC l (K )* . Hence, by Lemma 3.7, for

any f ∈ L∞ (K ),

*

UC l (K ) and as an element of L (K ) (see [14, §3, p. 13]).

Lemma 3.9. Let

m ∈ Z (UC l (K )* ) . By

prove that φ m ∈ Z (L∞ (K )* ) , by Lemma 2.3, it is

Z (UC l (K )* ) . □ 1

and

Remark 3.8, we may consider φ m in L1 (K )** . To

by the continuity of L m . Now by linearity, we have for

i

= 〈 m , π (n )f 〉 = 〈 m π (n ), f 〉.

each

〈 m n , f 〉 = 〈 n , fm 〉

m n = mn

i

= lim〈π (n )f , φi 〉 = lim〈φi , π (n )f 〉

l

where λi > 0 and Σiα=1λi = 1 such that nα → n in the w * -topology

i

= lim〈 n , f φi 〉 = lim〈π (n ), f φi 〉

in Z (UC l (K )* ) (see Remark 3.3)

i

fF is a continuous and mF of L∞ (K )* into itself.

*

θ ∈ C (K )* , then by (**), lim〈θ ,

J. Sci. I. R. Iran

π : L∞ (K )* → UC l (K )*

〈φ mnα , f 〉 = 〈φ (mnα ), f 〉 = 〈 mnα , f φ 〉

be

the

= 〈 m π (nα ), f φ 〉 → 〈 m π (n ), f φ 〉

adjoint of the inclusion map of UC l (K ) into L∞ (K ) .

= 〈φ (m π (n )), f 〉

Then π is w * - w * continuous and mn = m π (n ) for

= 〈φ m π (n ), f 〉 = 〈φ mn , f 〉 ,

each m , n ∈ L∞ (K )* .

46

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so by Lemma 2.3, φ m ∈ Z (L∞ (K )* ) .

Granirer for groups in [7, p. 62-64]. Another version of this Corollary was proved in [15, Theorem 19]. A.T. Lau has also proved it in [12, Corollary 4].

Conversely, let m ∈UC l (K )* , and nα → n in the w * -topology of UC l (K )* , then for each f ∈UC l (K ),

Corollary 3.15. Let K be a locally compact hypergroup. Then K is compact if and only if UC l (K ) = W AP (K ) .

there exists g ∈UC l (K ) and φ ∈ L1 (K ) such that f = g φ ([19, Lemma 2.2] and Lemma 2.4(ii)). Hence 〈 mnα , f 〉 = 〈 mnα , g φ 〉 = 〈φ (mnα ), g 〉

Proof. If K is compact, then by [8, 2.2D and 4.2F] we have UC l (K ) = C (K ) = W AP (K ) . For the converse,

= 〈φ mnα , g 〉 → 〈φ mn , g 〉 = 〈 mn , g φ 〉 = 〈 mn , f 〉.

from UC l (K ) = W AP (K ) = Z (UC l (K )* ) = M (K ) , it follows that K is compact. □ For the following corollary in the group case, see [12, Corollary 5].

Now we are ready for the main theorem of this section. Theorem 3.11. Z (UC l (K )* ) = M (K )

Corollary 3.16. Let K be a locally compact hypergroup. Then K is compact if and only if UC l (K ) has a unique left invariant mean.

Proof. By Lemmas 3.5 and 3.7, it is enough to show that Z (UC l (K )* ) ⊆ M (K ) . Let m ∈ Z (UC l (K )* ) , then by Lemma 3.2, m = μ + m1 , for some μ ∈ M (K )

Proof. If K is compact, then the normalized Haar measure is the unique left invariant mean on UC l (K ) = C (K ) . Conversely, let m be the unique left invariant mean on UC l (K ) , then one can check that mn is also left

⊥

and m1 ∈ C 0 (K ) . It is enough to show that m1 = 0 . Let φ ∈ L1 (K ) . Since C 0 (K ) ⊥ is an ideal in UC l (K )* (Lemma 3.2) φ m1 ∈ C 0 (K ) ⊥ and φ m1 ∈ Z (L1 (K )** ) , by Lemma 3.10. Hence φ m1 = 0 (Lemma 2.7). Let f ∈UC l (K ) , then f = g φ , for some g ∈UC l (K ) ,

invariant mean on UC l (K ) , for each n ∈UC l (K )* . Hence mn = λ m , for some complex number λ . Let {nα } be a net in UC l (K )* converging to n in the

and φ ∈ L1 (K ) ([19, Lemma 2.2] and Lemma 2.4(ii)), and

weak * -topology, and mnα = λα m , mn = λ m , then

λα = mnα (1) = nα (1) converges to n (1) = mn (1) = λ .

〈 m1 , f 〉 = 〈 m1 , g φ 〉 = 〈φ m1 , g 〉 = 〈φ m1 , g 〉 = 0.

Hence L m is weak * -weak * continuous, and by Theorem 3.11 and Proposition 3.7, m ∈ M (K ) and by [8, 7.2B], K is compact. □

Hence m1 = 0 , as desired. □ Corollary 3.12. If K is commutative, then M (K ) is

the algebraic center of UC l (K )* .

Acknowledgement

Corollary 3.13. Let m ∈UC l (K )* be such that L m is *

Vol. 18 No. 1 Winter 2007

The author would like to sincerely thank the referees for their valuable comments and useful suggestions.

*

weak -weak continuous on any bounded sphere of UC l (K )* , then m ∈ M (K ) .

References 1. Chilana A.K. Harmonic analysis and hypergroups. Proceedings of the Symposium on Recent Developments in Mathematics, Allahabad, pp. 93-121 (1978). 2. Dales H.G. and Lau A.T.-M. The second dual of Beurling algebras. Mem. Amer. Math. Soc., 177(836): (2005). 3. Duncan J. and Hosseiniun S.A.R. The second dual of a Banach algebra. Proc. Roy. Soc. Edinburgh, A84: 309325 (1979). 4. Dunkl C.F. The measure algebra of a locally compact hypergroup. Trans. Amer. Math. Soc., 179: 331-348

Definition 3.14. A bounded continuous function f is called weakly almost periodic if { x f : x ∈ K } is relatively weakly compact in the space of all bounded continuous functions on K . We denote the Banach space of all weakly almost periodic functions on K by W AP (K ) . The following corollary was proved by Skantharajah for hypergroups in [20, Theorem 4.2.7, p. 94], and by

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J. Sci. I. R. Iran

13. Lau A.T. and Losert V. On the second conjugate algebra of L1(G) of a locally compact group. J. London Math. Soc., 37(2): 464-470 (1988). 14. Lau A.T. and Ülger A. Topological centers of certain dual algebras. Trans. Amer. Math. Soc., 348: 1191-1212 (1996). 15. Medghalchi A.R. The second dual algebra of a hypergroup. Math. Z., 210: 615-624 (1992). 16. Pym J. S. Weakly separately continuous measure algebras. Math. Ann., 175: 207-219 (1968). 17. Ross K.A. Hypergroups and centers of measure algebras. Ist. Naz. di Alta. Mat., Symposia Math., 22: 189-203 (1977). 18. Ross K.A. Signed hypergroups- a survey. Applications of Hypergroups and Related Measure Algebras. Contemporary Math., 22: 319-329 (1995). 19. Skantharajah M. Amenable hypergroups. Ill. J. Math., 36: 15-46 (1992). 20. Skantharajah M. Amenable hypergroups. Doctoral Thesis, University of Alberta (1989). 21. Spector R. Apercu de la theorie des hypergroupes, Analyse Harmonique sur les groupes de Lie. Lecture Notes in Math., 497, Springer-Verlag, Berlin, Heidelberg, New York (1975).

(1973). 5. Ghaffari A. and Metghalchi A.R. Acta Mathematica Sinica, English series, 20(2): 201-208 (2004). 6. Ghahramani F., Lau A.T., and Losert V. Isometric, isomorphism between Banach algebras related to locally compact groups. Trans. Amer. Math. Soc., 321: 273-283 (1990). 7. Granirer E.E. Exposed points of convex sets and weak sequential convergence. Mem. Amer. Math. Soc., 123 (1972). 8. Jewett R.I. Spaces with an abstract convolution of measures. Advances in Math., 18: 1-101 (1975). 9. Kamyabi-Gol R.A. The Topological center of L1(K)**. Scientiae Mathematicae Japonicae, 62(1): 81-89 (2005). 10. Kamyabi-Gol R.A. A short proof for the existence of Haar measure on commutative hypergroups. Journal of Sciences, Islamic Republic of Iran, 13(3): 263-265 (2002). 11. Kelly J.L. General Topology. Van Nostrand, New York (1968). 12. Lau A.T. Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups. Math. Proc. Camb. Phil. Soc., 99: 273-283 (1986).

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