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Strong Arens irregularity of Beaurling algebras with locally conex topology

Authors:
Saeid Maghsoudi at University of Zanjan
Saeid Maghsoudi
Rasoul Nasr-Isfahani at Isfahan University of Technology
Rasoul Nasr-Isfahani
Ali Rejali at University of Isfahan
Ali Rejali
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Abstract

Let G be a locally compact group with a weight function ω. Recently, we have shown that the Banach space L 0∞ (G,1/ω) can be identified with the strong dual of L 1(G, ω)equipped with some locally convex topologies τ. Here we use this duality to introduce an Arens multiplication on (L 1(G, ω), τ)**, and prove that the topological center of (L 1(G, ω), τ)** is (L 1(G, ω); this enables us to conclude that (L 1(G, ω), τ) is Arens regular if and only if G is discrete. We also give a characterization for Arens regularity of L 0∞ (G, 1/ω)1.
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Arch. Math. 86 (2006) 437–448
0003–889X/06/050437–12
DOI 10.1007/s00013-005-1496-6
© Birkh¨
auser Verlag, Basel, 2006 Archiv der Mathematik
Strong Arens irregularity of Beurling
algebras with a locally convex topology
By
S. Maghsoudi, R. Nasr-Isfahani and A. Rejali
Abstract. Let Gbe a locally compact group with a weight function ω. Recently, we have shown
that the Banach space L∞
0(G, 1/ω) can be identified with the strong dual of L1(G, ω) equipped with
some locally convex topologies τ. Here we use this duality to introduce an Arens multiplication
on (L1(G, ω), τ )∗∗, and prove that the topological center of (L1(G, ω), τ )∗∗ is L1(G, ω); this
enables us to conclude that (L1(G, ω), τ ) is Arens regular if and only if Gis discrete. We also
give a characterization for Arens regularity of L∞
0(G, 1/ω)1.
1. Introduction. Throughout this paper, Gdenotes a locally compact group with a
fixed left Haar measure λ. We also assume that ωbeaweight function on G, that is a
continuous function ω:G→[1,∞)with
ω(xy)ω(x) ω(y) (x,y ∈G).
The Beurling algebra L1(G, ω) is defined to be the space of all measurable functions ϕ
such that ωϕ ∈L1(G), the group algebra of Gas defined in [12]. Then L1(G, ω) with the
convolution product ∗and the norm .1,ω defined by
ϕ1,ω =ωϕ1(ϕ ∈L1(G, ω))
is a Banach algebra. We follow Dales and Lau [4] in our definitions and notations for
Beurling algebras.
Also, let L∞(G, 1/ω) denote the space of all measurable functions fwith f/ω ∈L∞(G),
the Lebesgue space as defined in [12]. Then L∞(G, 1/ω) with the product .ωdefined by
f.ωg=fg/ω (f, g ∈L∞(G, 1/ω)),
the norm .∞,ω defined by
f∞,ω =f/ω∞(f ∈L∞(G, 1/ω)),
Mathematics Subject Classification (2000): Primary 43A20, 46H05; Secondary 43A10, 43A15, 46A03.
438 S. Maghsoudi, R. Nasr-Isfahani and A. Rejali arch. math.
and the complex conjugation as involution is a commutative C∗-algebra. Moreover,
L∞(G, 1/ω) is the dual of L1(G, ω). In fact, the mapping Tfrom L∞(G, 1/ω) to
L1(G, ω)∗defined by
T(f),ϕ=
G
f(x) ϕ(x) dλ(x)
is an isometric isomorphism.
We denote by L∞
0(G, 1/ω) the C∗-subalgebra of L∞(G, 1/ω) consisting of all functions
f∈L∞(G, 1/ω) that vanish at infinity; i.e., for each ε>0, there is a compact subset
Kof Gfor which
fχ
G\K∞,ω <ε.
The problem of the topological center for (L1(G, ω), .1,ω)∗∗ has been studied by
several authors; see Baker and Rejali [1], Craw and Young [3], Dzinotyiweyi [5], and
Neufang [10]. For a nice survey on the subject see Dales and Lau [4]. The recent works
of Neufang [10] and Dales and Lau [4] show the weight function plays a significant role in
the study of the topological center for (L1(G, ω), .1,ω)∗∗ .
On the other hand, it has been shown in [8] that the Banach space L∞
0(G, 1/ω) can be
identified with the strong dual of L1(G, ω) equipped with certain locally convex topologies.
It also has been proved that, except for the case where Gis finite, there are infinitely many
such locally convex topologies τon L1(G, ω).
In this paper, we use this duality to introduce an Arens multiplication on (L1(G, ω), τ)∗∗,
and prove that L1(G, ω) is the topological center of (L1(G, ω), τ )∗∗; this enables us to
conclude that (L1(G, ω), τ ) is Arens regular if and only if Gis discrete. Our results
improve some interesting results obtained in Singh [13] on group algebras to the weighted
case.
2. Preliminaries. We commence this section with the following lemma. For each
function ϕon G, we denote by ˜ϕthe function defined on Gby
˜ϕ(x) =ϕ(x−1)(x∈G).
We also denote by the modular function on G, and by C0(G, 1/ω) the space of all
continuous functions fon Gsuch that f/ω vanish at infinity.
Lemma 2.1. Fo r ϕ∈L1(G, ω) and g∈L∞
0(G, 1/ω) we have
1
˜ϕ∗g, g ∗˜ϕ∈C0(G, 1/ω).
Proof. Let ε>0 be given and choose a compact subset Cof Gwith
gχ
G\C∞,ω <ε.
Vol. 86, 2006 Strong Arens irregularity of Beurling algebras 439
Now, let ϕ∈L1(G, ω). First, note that 1
˜ϕ∗gis continuous on Gby 3.7.9 of [12].
Now, let Dbe a compact subset of Gwith
ϕχ
G\D1,ω <ε.
Then for each x∈G\D−1Cwe get Dx G\Cand hence
1
˜ϕ∗g(x)
=
G
(y−1)ϕ(y
−1)g(y
−1x) d λ(y)
=
G
ϕ(y) g(yx) dλ(y)
ω(x)
G
ω(y) |ϕ(y)||g(yx)|
ω(yx) dλ(y)
ω(x)
G\D
|(ωϕ)(y )||(g/ω)(yx )|dλ(y)
+
D
|(ωϕ)(y )||(g/ω)(yx )|dλ(y)
ω(x) (ϕχ
G\D1,ω g∞,ω +ϕ1,ω gχ
G\C∞,ω)
It follows that for every x∈G\D−1C,
1
˜ϕ∗g(x)
<ε(g∞,ω +ϕ1,ω ) ω (x).
Since D−1Cis compact in G, this means that (1
˜ϕ∗g)/ω vanishes at infinity, and so
1
˜ϕ∗g∈C0(G, 1/ω).
Similarly g∗˜ϕ∈C0(G, 1/ω).
By M(G,ω), we mean the space of all complex regular Borel measures µon Gsuch
that ωµ ∈M(G), the measure algebra of Gwith the convolution product ∗and the total
variation norm .defined as in [12]. Then M(G, ω) with the product ∗and the norm .ω
defined by
µω=ωµ(µ ∈M(G, ω))
is a Banach algebra. Moreover, M(G, ω) is the dual of C0(G, 1/ω) for the pairing
µ, f =
G
f (x) dµ(x)
for all µ∈M(G, ω) and f∈C0(G, 1/ω); see [4] for details.
440 S. Maghsoudi, R. Nasr-Isfahani and A. Rejali arch. math.
For every m∈L∞
0(G, 1/ω)∗and g∈L∞
0(G, 1/ω), we denote by mg the function in
L∞(G, 1/ω) defined by
T (mg), ϕ=m, 1
˜ϕ∗g(ϕ ∈L1(G, ω)).
Proposition 2.2. The space L∞
0(G, 1/ω) is left introverted in L∞(G, 1/ω); i.e. for each
m∈L∞
0(G, 1/ω)∗and g∈L∞
0(G, 1/ω), we have mg ∈L∞
0(G, 1/ω).
This result lets us to endow L∞
0(G, 1/ω)∗with the first Arens product ·defined by
m·n, g=m, ng
for all m, n ∈L∞
0(G, 1/ω)∗and g∈L∞
0(G, 1/ω). Then L∞
0(G, 1/ω)∗with this product
is a Banach algebra.
Proof of Proposition 2.2. Let εbe given and choose a compact subset Cof
Gwith |g(t)|<εω(t)for locally almost all t∈G\C.
Now, let m∈L∞
0(G, 1/ω)∗. Since mg ∈L∞(G, 1/ω), we only need to prove that mg
vanishes at infinity.
To that end, we may assume that gis real-valued and non-negative. Also, since
L∞
0(G, 1/ω)∗is spanned by its positive elements, we can assume m0. Let σ∈M(G,ω)
denote the restriction of mto C0(G, 1/ω). Then for every ε>0, there is a compact subset
Eof Gsuch that
(ωσ )(G \E) < ε
2.
Let {Eα}be the family of compact subsets of Gdirected by upward inclusion. Then
(ω χEα)is a bounded approximate identity for the C∗-algebra L∞
0(G, 1/ω). Now, let nbe
the linear functional on L∞
0(G, 1/ω) defined by
n, h=m, h χG\E
for all h∈L∞
0(G, 1/ω). Since nis a positive functional on L∞
0(G, 1/ω), it follows that
n=limαn, ω χEα. So there exists α0such that
n, ω χEα0n− ε
2.
Choose a function φ∈Cc(G) with χEα0φ1. Then
n, ω χEα0n, ω φ
n|C0(G,1/ω)
=(ωσ )(G \E)
which shows that n<ε.
Vol. 86, 2006 Strong Arens irregularity of Beurling algebras 441
For each positive function ϕ∈L1(G, ω) with supp(ϕ) G\CE−1, there is a compact
subset Dof Gfor which
DG\CE−1and ϕχ
G\D1,ω <ε.
Since D−1C∩E=∅, it follows from the proof of Lemma 2.1 that
1
˜ϕ∗gχE
∞,ω
<ε(g∞,ω +1)
and therefore
G\CE−1
(mg)(x) ϕ (x) dλ(x ) =T (mg), ϕ
=m, 1
˜ϕ∗g
=m, 1
˜ϕ∗gχE+n, 1
˜ϕ∗g
ε(g∞,ω +1)m+εg∞,ω
This shows that
(mg) χG\CE−1∞,ω ε[(g∞,ω +1)m+g∞,ω].
That is mg ∈L∞
0(G, 1/ω).
We end this section by the following characterization of Arens regularity of L∞
0(G, 1/ω)∗.
First, let us recall that L∞
0(G, 1/ω)∗is called Arens regular if the map n−→ m·nis weak∗-
weak∗continuous on L∞
0(G, 1/ω)∗for all m∈L∞
0(G, 1/ω)∗.
Also, let be the function defined on G×Gby
(x,y) =ω (xy )/ω(x)ω (y)
for all x,y ∈G. Then is called zero cluster if
lim
nlim
m(xn,y
m)=0=lim
mlim
n(xn,y
m)
for all sequences {xn}and {yn}in Gwith distinct elements whenever both iterated limits of
(xn,y
m)exists.
Theorem 2.3. The Banach algebra L∞
0(G, 1/ω)∗is Arens regular if and only if Gis
finite, or discrete and is zero cluster.
Proof. Suppose that L∞
0(G, 1/ω)∗is Arens regular. Since L1(G, ω) is a closed sub-
algebra of L∞
0(G, 1/ω)∗, it follows from [2] that L1(G, ω) is also Arens regular. So we
only need to recall from [3] that L1(G, ω) is Arens regular if and only if Gis finite, or
discrete and is zero cluster; see also [1] and [4]. The converse follows from the fact that
L∞
0(G, 1/ω)∗=1(G, ω) if Gis discrete.
442 S. Maghsoudi, R. Nasr-Isfahani and A. Rejali arch. math.
3. Strong Arens irregularity of L1(G, ω)
L1(G, ω)
L1(G, ω).Let Cdenote the set of increasing sequences
of compact subsets of G, and Rdenote the set of increasing sequences (rn)of real numbers
in (0,∞)with rn→∞. For any (Cn)∈Cand (rn)∈R, set
U((Cn), (rn)) ={ϕ∈L1(G, ω) :ϕχ
Cn1,ω rnfor all n1},
and note that U((Cn), (rn)) is a convex balanced absorbing set in the space L1(G, ω).It
is easy to see that the family Uof all sets U((Cn), (rn)) for (Cn)∈Cand (rn)∈R,isa
base of neighbourhoods of zero for a locally convex topology on L1(G, ω). We denote this
topology by β1(G, ω).
Let us recall from [8] that the strong dual of (L1(G, ω), β1(G, ω)) can be identified with
the Banach space (L∞
0(G, 1/ω), .∞,ω). In fact, the restriction T0of Tto L∞
0(G, 1/ω),
is a continuous isomorphism from L∞
0(G, 1/ω) onto (L1(G, ω), β1(G, ω))∗. Thus the
adjoint T∗
0of T0is an identification between (L1(G, ω), β1(G, ω))∗∗ and L∞
0(G, 1/ω)∗.
We denote by σ0(G, ω) the weak topology σ(L
1(G, ω), L∞
0(G, 1/ω)) on L1(G, ω). Let
us remark that L∞
0(G, 1/ω) is the dual of (L1(G, ω), τ ) for all locally convex topology τ
on L1(G, ω) with
σ0(G, ω) τβ1(G, ω),
and recall from [8] that there are infinitely many such locally convex topologies τon
L1(G, ω) if Gis infinite.
Theorem 3.1. Let τbe a locally convex topology on L1(G, ω) such that
σ0(G, ω) τβ1(G, ω). Then (L1(G, ω), τ )∗∗ with the first Arens product can be
identified with a Banach algebra, where FHis defined by the equation
T∗
0(F H) =T∗
0(F ) ·T∗
0(H )
for all F,H ∈(L1(G, ω), τ )∗∗.
Proof. The map T∗
0is an identification between (L1(G, ω), τ )∗∗ and the Banach space
L∞
0(G, 1/ω)∗. Moreover, for any F,H ∈(L1(G, ω), τ )∗∗ we have T∗
0(F H) ∈
L∞
0(G, 1/ω)∗from which it follows that
FH∈(L1(G, ω), τ )∗∗.
So is well-defined on (L1(G, ω), τ )∗∗. It is easy to see that defines an algebra product
on (L1(G, ω), τ )∗∗ and that T∗
0is also an algebra isomorphism.
Proposition 3.2. Let τbe a locally convex topology on L1(G, ω) such that σ0(G, ω) τ
β1(G, ω). Then (L1(G, ω), τ ) is a closed ideal in its second dual equipped with the
strong topology.
Proof. Since T∗
0(L1(G, ω)) is a closed subspace of L∞
0(G, 1/ω)∗, it follows that
(L1(G, ω), τ ) is a closed subspace of (L1(G, ω), τ )∗∗.
Vol. 86, 2006 Strong Arens irregularity of Beurling algebras 443
Now, suppose that ϕ∈L1(G, ω) and F∈(L1(G, ω), τ )∗∗. We show that ϕF∈
L1(G, ω); that Fϕ∈L1(G, ω) is similar. Set
m:=T∗
0(F ) ∈L∞
0(G, 1/ω)∗
and let ν∈M(G, ω) be the restriction of mto C0(G, 1/ω). Then ϕ∗ν∈L1(G, ω). So, it
suffices to show that ϕF=ϕ∗ν.
To that end, note that if f∈(L1(G, ω), τ )∗, then
g=T−1
0(f ) ∈L∞
0(G, ω),
and hence
1
˜ϕ∗g∈C0(G, 1/ω)
by Lemma 2.1. Therefore
m, 1
˜ϕ∗g=ν, 1
˜ϕ∗g.
Now, on the one hand,
ϕF,f=T∗
0(ϕ F),g
=T∗
0(ϕ), mg
=ϕ, T0(mg)
=m, 1
˜ϕ∗g,
and on the other hand,
ν, 1
˜ϕ∗g=
G
1
˜ϕ∗g(y)dν(y)
=
G
G
g(xy)ϕ(x)dλ(x)dν(y)
=
G
g(t) d(ϕ ∗ν)(t )
=ϕ∗ν, f .
That is ϕF=ϕ∗νas required.
Let τbe as in Theorem 3.1. For any Hin (L1(G, ω), τ )∗∗, the map F−→ FHis
weak∗-weak∗continuous on (L1(G, ω), τ )∗∗. For an element Fin (L1(G, ω), τ )∗∗, the
map H−→ FHis in general not weak∗-weak∗continuous on (L1(G, ω), τ )∗∗ unless
Fis in L1(G, ω).
444 S. Maghsoudi, R. Nasr-Isfahani and A. Rejali arch. math.
The topological center of (L1(G, ω), τ )∗∗ with respect to is denoted by
Z1((L1(G, ω), τ )∗∗)and is defined to be the set of all F∈(L1(G, ω), τ )∗∗ for which
the map H−→ FHis weak∗-weak∗continuous on (L1(G, ω), τ )∗∗. We say that
(L1(G, ω), τ ) is strongly Arens irregular if
Z1((L1(G, ω), τ )∗∗)=L1(G, ω).
We are now ready to prove our main result of this paper.
Theorem 3.3. Let τbe a locally convex topology on L1(G, ω) such that σ0(G, ω)
τβ1(G, ω). Then (L1(G, ω), τ ) is strongly Arens irregular.
Proof. Suppose that F∈Z1((L1(G, ω), τ )∗∗), and let ν∈M(G, ω) be the restriction
of T∗
0(F ) to C0(G, 1/ω). Since C0(G, 1/ω) is weak∗dense in L∞
0(G, 1/ω), to prove that
F∈L1(G, ω), we only need to show that ων is absolutely continuous with respect to λ.
By Lemma 1.1 of [5], ωis bounded on compact subset of G; so the result will follow if we
show that νis absolutely continuous with respect to λ.
To that end, it suffices to prove that for any compact subset Cof G, the Borel measurable
function h:x−→ ν(Cx−1)is equal locally almost every where to a continuous function
on G; see [11].
Let Cbe a compact subset of Gand set g=χC. Then for every ϕ∈L1(G, ω), the
function g∗˜ϕdefined on Gbelongs to C0(G, 1/ω) by Lemma 2.2. Thus
ϕ, T0(h)=
G
ν(Cx−1)ϕ(x) d λ(x)
=
G
G
χCx−1(y)ϕ(x) d ν(y) d λ(x)
=
G
G
g(yx) ϕ(x) dλ(x) dν(y)
=
G
(g ∗˜ϕ)(y) dν(y)
=T∗
0(F ), g ∗˜ϕ.
Since g∗˜ϕ=T∗
0(ϕ)g, it follows that
ϕ, T0(h)=Fϕ, T0(g).
Now, let H∈(L1(G, ω), τ )∗∗ and choose a net (ϕγ)in L1(G, ω) such that ϕγ−→ H
in the weak∗topology of (L1(G, ω), τ )∗∗. Since the map G−→ FGis weak∗-weak∗
continuous on (L1(G, ω), τ )∗∗ we have
H,T0(h)=lim
γϕγ,T
0(h)
=lim
γFϕγ,T
0(g)
=FH, T0(g).
Vol. 86, 2006 Strong Arens irregularity of Beurling algebras 445
Next, let (ψα)be a net in L1(G, ω) with ψα−→ Fin the weak∗topology of (L1(G, ω), τ )∗∗.
Then
FH,T0(g)=lim
αψαH,T0(g)
=lim
αT∗
0(ψα), T ∗
0(H ) g
=lim
αψα,T∗
0(H ) g
=lim
αT∗
0(H ), 1
˜
ψα∗g
from which it follows that
T∗
0(H ), h=lim
αT∗
0(H ), 1
˜
ψα∗g.
Since elements of L∞
0(G, 1/ω)∗are of the form T∗
0(H ) for some Hin the second dual of
(L1(G, ω), τ ), it follows that
1
˜
ψα∗g−→ h
in the weak topology of L∞
0(G, 1/ω). According to Lemma 2.2,
1
˜
ψα∗g∈C0(G, 1/ω)
for all α. Since C0(G, 1/ω) is weakly closed in L∞
0(G, 1/ω) we conclude that his identical
to a function in C0(G, 1/ω).
Let τbe as in Theorem 3.1. The algebra (L1(G, ω), τ ) is called Arens regular if the map
G−→ FGis weak∗-weak∗continuous on (L1(G, ω), τ )∗∗ for all F∈(L1(G, ω), τ)∗∗ ;
i.e.,
Z1((L1(G, ω), τ )∗∗)=(L1(G, ω), τ )∗∗.
Let us also recall that (L1(G, ω), τ ) is called semireflexive if
L1(G, ω) =(L1(G, ω), τ )∗∗.
As a consequence of the above theorem, we have the following description of Arens
regularity of (L1(G, ω), τ ).
Corollary 3.4. Let τbe a locally convex topology on L1(G, ω) such that σ0(G, ω) τ
β1(G, ω). Then the following assertions are equivalent.
(a) (L1(G, ω), τ ) is Arens regular.
(b) (L1(G, ω), τ ) is semireflexive.
(c) Gis discrete.
446 S. Maghsoudi, R. Nasr-Isfahani and A. Rejali arch. math.
Proof. Wefirst show that (c) implies (a). Suppose that Gis discrete. Then L1(G, ω)
is the space 1(G, ω) of all functions ϕon Gsuch that
x∈G
|ϕ(x)|ω(x) < ∞. Since the
dual of (1(G, ω), τ ) equipped with the strong topology can be identified with the Banach
space ∞
0(G, 1/ω) equipped with the .∞,ω-topology, where ∞
0(G, 1/ω) denotes the
space of all bounded complex-valued functions fon Gsuch that f/ω vanishes at infinity.
Furthermore, ∞
0(G, 1/ω)∗can be identified with 1(G, ω). Thus
(1(G, ω), τ )∗∗ =1(G, ω).
In particular,
Z1((1(G, ω), τ )∗∗)=(1(G, ω), τ )∗∗
by the definition of topological center. That is (1(G, ω), τ ) is Arens regular.
Also, (a) implies (b) by Theorem 3.3.
We prove that (b) implies (c). For this end, suppose that (L1(G, ω), τ ) is semireflexive.
Then
T∗
0((L1(G, ω), τ )∗∗)=T∗
0((L1(G, ω), τ )).
By Theorem 3.3 we have L∞
0(G, 1/ω)∗=T∗
0((L1(G, ω), τ )). Let Ebe an extension of δe
from C0(G, 1/ω) to an element of L∞
0(G, 1/ω)∗, where edenotes the identity element of
G. Then E=T∗
0(ϕ) for some ϕ∈L1(G, ω), and m·E=mfor all m∈L∞
0(G, 1/ω)∗.
Therefore
T∗
0(ϕ F) =T∗
0(F ) ·T∗
0(ϕ) =T∗
0(F ) ·E=T∗
0(F )
for all F∈(L1(G, ω))∗∗. In particular, ϕ∗ψ=ψfor all ψ∈L1(G, ω). That is, L1(G, ω)
has an identity, and hence (c) holds; see [4].
Corollary 3.5. Let τbe a locally convex topology on L1(G, ω) such that σ0(G, ω) τ
β1(G, ω).If(L1(G, ω), .1,ω )is Arens regular, then so is (L1(G, ω), τ ).
Proof. Recall from [3] that Gis discrete if (L1(G, ω), .1,ω )is Arens regular, and
hence the result follows from Corollary 3.4.
We conclude this work by some examples which show that the converse of Corollary 2.5
is not valid.
Example 3.6. (a) Let ω(m, n) =(1+|m|)(1+|n|)for all m, n ∈Z. Then ωis a
weight function on the additive discrete group Z2. By Theorem 3.3 and its Corollary 3.4
we have
1(Z2,ω) =Z1((1(Z2,ω),τ)
∗∗)=(1(Z2,ω),τ)
∗∗
for any locally convex topology τon 1(Z2,ω) with
σ0(Z2,ω)τβ1(Z2,ω);
Vol. 86, 2006 Strong Arens irregularity of Beurling algebras 447
whereas Example 8.4 of [4] shows that
1(Z2,ω) Z1((1(Z2,ω),.1,ω )∗∗)(1(Z2,ω),.1,ω )∗∗.
(b) For each α>0, let ωα(x) =(1+|x|)αfor all x∈R. Then ωαis a weight function
on the additive locally compact group Rwith usual topology. It follows from Theorem 3.3
that (L1(R,ω
α), τ ) is strongly Arens irregular and from Corollary 3.4 that (L1(R,ω
α), τ )
is not Arens regular, where τis a locally convex topology on L1(R,ω
α)with
σ0(R,ω
α)τβ1(R,ω
α).
However, Theorem 11.6 of [4] implies that (L1(R, ωα), .1,ω )is neither strongly Arens
irregular nor Arens regular.
Remark 3.7. It is well-known that (L1(G), .1)is strongly Arens irregular, from
which it follows that (L1(G), .1)is Arens regular if and only if Gis finite. This result
is due to Isik, Pym, and Ulger [6] for a compact group G, and Lau and Losert [7] for a
locally compact group G; see also Neufang [9] for a totally different proof, and Young [14].
A special case of our results conclude that if Gis compact, then L1(G) is strongly Arens
irregular; in particular, L1(G) is Arens regular if and only if Gis finite.
Acknowledgment. The authors would like to thank the referee of the paper for
invaluable comments. This research was supported by the center of Excellence for Mathe-
matics at the Isfahan University of Technology and the University of Isfahan.
References
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Received: 8 March 2005
S. Maghsoudi and A. Rejali R. Nasr-Isfahani
Department of Mathematics Department of Mathematics
University of Isfahan Isfahan University of Technology
Isfahan Isfahan
Iran Iran
maghsodi@sci.ui.ac.ir isfahani@cc.iut.ac.ir
rejali@sci.ui.ac.ir
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