Introverted subspaces of the duals of measure algebras

0.17 MB·

by  H. Javanshiri

Introverted subspaces of the duals of measure algebras
7
H. Javanshiri and R. Nasr-Isfahani
1
0
2

n Abstract
a
J Let G be a locally compact group. In continuation of our studies on the first and second

6 dualsofmeasurealgebrasbytheuseofthetheoryofgeneralisedfunctions,herewestudythe
2 ∗ ∗
C -subalgebra GL0(G) of GL(G) as an introverted subspace of M(G) . In the case where G
⊥
] is non-compactwe showthat anytopologicalleftinvariantmeanonGL(G) lies inGL0(G) .
A ∗
We then endow GL0(G) with an Arens-type product which contains M(G) as a closed sub-
F
algebra and Ma(G) as a closed ideal which is a solid set with respect to absolute continuity
h. in GL0(G)∗. Among other things, we prove that G is compact if and only if GL0(G)∗ has a
t
a non-zero left (weakly) completely continuous element.
m
[ Mathematics Subject Classification (2010). 43A10, 43A15, 43A20, 47B07.

Key words. Measure algebra, generalised functions vanishing at infinity, introverted sub-
1
v space, topological invariant mean, completely continuous element.
1
5
7 1 Introduction
7
0
Allover thispaperG isalocally compact groupwithleftHaar measureλandidentity element e,
.
1
and the notations C (G) and C (G) refer to the space of all functions with compact supportand
0 c 0
7 the space of all functions vanishing at infinity, respectively. Moreover, the letter M(G) means
1
that the measure algebra of G consisting of all complex regular Borel measures on G with the
:
v
total variation norm and the convolution product “∗” defined by the formula
i
X
r hµ∗ν,gi = g(xy) dµ(x) dν(y)
a Z Z
G G
for all µ,ν ∈ M(G) and g ∈ C (G). It is folklore that M(G) is the first dual space of C (G) for
0 0
the pairing
hµ,gi := g(x) dµ(x) µ ∈M(G), g ∈ C (G) .
ZG (cid:16) 0 (cid:17)
In the last thirty years, research on the second duals of Banach algebras has mostly centred
around the Banach algebras related to locally compact groups, and has been dealt with by Lau
et al. in the works [3, 4, 7, 11]. In particular, [7] is the first important work devoted to the
study of the second duals of measure algebras. Among other things, the authors of [7] have
conjectured that the Banach algebra M(G) is strongly Arens irregular, and its second dual,
M(G)∗∗, determines G in the category of all locally compact groups. Later on, the second duals
of the measure algebras has been studied in a series of papers. In particular, [4] is the second
important work devoted to the study of M(G)∗∗, where most of the known results about these
1
2 H. Javanshiri and R. Nasr-Isfahani
Banach algebras up to the year 2012 can be found in it. Recall from [4] that M(G)∗, the first
dual space of M(G), as the second dual of the C∗-algebra C (G) is a commutative unital C∗-
0
algebra, and therefore if G denotes the hyper-Stonean envelope of G, then we can recognizing
M(G)∗ as C(G), the spaceeof all bounded complex-valued continuous functions on G. It follows
that M(G)∗∗ ∼=e M(G), where ∼= denotes the isometric algebra isomorphism and maeny authors,
up to the year 2012e, have used a type of this identification as a tool for the study of M(G)∗∗,
see [4] and the references therein for more details.
Recently intheworks[6,10],westudiedthefirstandseconddualsofmeasurealgebrasbythe
useof the theory of generalised functions which have been introduced and investigated by S˘re˘idr
[14] and Wong [15, 16]. In those papers we observed that GL (G), the space of all generalised
0
functions that vanishes at infinity, plays a crucial role in our investigation. Motivated by this,
here we study the C∗-algebra GL (G) as an introverted subspace of M(G)∗. In particular, in the
0
case where G is non-compact we show that any topological left invariant mean on GL(G) lies
in GL (G)⊥ which demonstrates that the weak∗-closed subspace GL (G)⊥ of M(G)∗ is far from
0 0
devoid of interest. We then endow GL (G)∗ with an Arens-type product which contains M(G)
0
and M (G) as a closed subalgebra and a closed ideal, respectively. Among other things, we
a
prove that the existence of a non-zero left (weakly) completely continuous element in GL (G)∗
0
is equivalent to the compactness of G.
2 Generalised functions: an overview
In this section, we give a brief overview of generalised functions in the sense of Wong [15].
Nevertheless, we shall require some facts about the theory of C∗-algebra. For background on
thistheory,weuse[13]asareferenceandadoptthatbook’snotation. Moreover,ournotationand
terminology are standard and, concerning Banach algebras related to locally compact groups,
they are in general those of the book [9] of Hewitt and Ross. The reader will remark that this
section is mostly taken from the papers by Wong [15, 16]; this is because of, we actually need
this section for the convenience of citation and a better exposition.
For any complex regular Borel measure µ on G, let L∞(|µ|) denote the Banach space of
all essentially bounded |µ|–measurable complex functions f on G with the essential supremum
µ
norm
kf k = inf α ≥ 0: |f | ≤ α, |µ|−a.e. .
µ µ,∞ µ
n o
Consider the product linear space {L∞(|µ|) : µ ∈ M(G)}. An element f = (f ) in this
µ µ∈M(G)
product is called a generalised funcQtion if f = f |µ|−a.e. for any µ,ν ∈ M(G) with µ ≪ ν,
µ ν
where µ ≪ ν means that |µ| is absolutely continuous with respect to |ν|. We note that this
condition implies that for given generalised functions f = (f )
µ µ∈M(G)
sup kf k :µ ∈ M(G) < ∞;
µ µ,∞
n o
otherwise, there is a sequence (µ ) in M(G) for which kf k ≥ n for all n ∈ N. Set
n µn µn,∞
µ = ∞ 2−nkµ k−1|µ |. Then µ ≪ µ, and hence kf k ≥ kf k ≥ n for all n ∈ N
n=1 n n n µ µ,∞ µn µn,∞
whichPis a contradiction.
Second dual of measure algebra 3
Now, following Wong [15] we use the notation GL(G) to denote the commutative unital C∗-
algebra of all generalised functions endowed with the coordinatewise operations, the involution
f 7→ f∗, where f∗ := (f ) , and norm
µ µ∈M(G)
kfk := sup kf k : µ ∈M(G) ,
∞ µ µ,∞
n o
where f = (f ) is in GL(G). The identity element of GL(G) is of course the generalised
µ µ∈M(G)
function 1 := (1 ) , where 1 is the identity element of L∞(|µ|). Moreover, we write
µ µ∈M(G) µ
f = (f ) ≥ 0 to mean that the generalised function f is positive in the C∗-algebra sense
µ µ∈M(G)
and denote by GL(G)+ the set of all positive elements of GL(G).
Remark 2.1 It is not hard to check that a generalised function f = (f ) is positive in
µ µ∈M(G)
the C∗-algebra sense if and only if f (x) ≥ 0 for all x ∈ G and all µ ∈ M(G); see [13, Page. 45]
µ
and [15, Page. 85] for more information.
As a main result, Wong [15] has shown that for each f = (f ) in GL(G), the equation
µ µ∈M(G)
hΨ(f),ζi:= f (x) dζ(x) (ζ ∈ M(G))
Z ζ
G
defines a linear functional Ψ(f) on M(G). In particular, the map f 7→ Ψ(f) is an isometric
linear mapping from GL(G) onto M(G)∗; see [15, Theorem 2.1 and Theorem 2.2] and [14] for
the same result in the special case where G is a certain locally compact abelian group. In
particular, any L ∈ M(G)∗ can be considered as a generalised function Ψ−1(L), and we do
not distinguish between a generalised function f and its unique corresponding linear functional
Ψ(f). In particular, this duality allows us to consider GL(G) as a Banach M(G)-bimodule. In
details, if ζ and f = (f ) are arbitrary elements of M(G) and GL(G), respectively. Then,
µ µ∈M(G)
one can considered the linear functionals fζ and ζf on M(G) defined by
hfζ,µi= hf,ζ ∗µi, hζf,µi= hf,µ∗ζi µ ∈ M(G) .
(cid:16) (cid:17)
In order to find the generalised functions corresponding to these linear functionals, following
Wong [15] and [16, Page 610], we define ζ ◦f ∈ {L∞(|µ|) : µ ∈ M(G)} as
Q
(ζ ◦f) = l f (µ ∈ M(G)),
µ ζ ζ∗µ
where
l f (y) = f (xy) dζ(x) for |µ|−a.e y ∈ G.
ζ ζ∗µ Z ζ∗µ
G
Then ζ ◦f is again a generalised function such that
hζ ◦f,µi= hf,ζ ∗µi (µ ∈ M(G));
see pages 88 and 89 of [15]. So, ζ◦f is the generalised function corresponding to the functional
fζ ∈ M(G)∗ such that Ψ(ζ ◦f) = fζ. Also, by using the right convolution notation, we can
show that f◦ζ is the generalised function correspondingto the functional ζf ∈ M(G)∗. In what
follows, we do not distinguish between the linear functionals fζ and ζf and their corresponding
generalised functions.
Later we will need the following remark in our present investigation.
4 H. Javanshiri and R. Nasr-Isfahani
Remark 2.2 Suppose that BM(G) denote the Banach space of all bounded Borel measurable
functions on G with the supremum norm k · k . Then each f ∈ BM(G) may be regarded
u
as an element (f ) in GL(G), where for each µ ∈ M(G) the functions f denotes the
µ µ∈M(G) µ
equivalent class of f in L∞(|µ|). Hence BM(G) can beconsidered as a closed subspaceof GL(G)
containing thespace C (G) of all complex-valued continuous boundedfunctions on G. Moreover,
b
each f ∈ BM(G) may be regarded as an element in M(G)∗ by the pairing hf,µi = f dµ
G
(µ ∈ M(G)). In this case, the restriction of the map Ψ to BM(G) is precisely the embedRding of
BM(G) into M(G)∗.
3 Generalised functions that vanish at infinity
We commence this section by recalling the main object of the work which is introduced and
studied by the authors in [10].
Definition 3.1 A generalised function f = (f ) vanishes at infinity if for each ε > 0,
µ µ∈M(G)
there is a compact subset K of G for which kf χ k < ε for all µ ∈ M(G); formally
ε µ G\Kε µ,∞
∀ε >0 ∃K ∈K(G) s.t. ∀µ ∈ M(G), |f (x)| < ε for |µ|−almost all x ∈ G \K ,
ε µ ε
where χ denotes the characteristic function of K on G and K(G) denotes the setof allcompact
Kε ε
subsets in G.
We denote by GL (G) the C∗-subalgebra of GL(G) consisting of all generalised functions
0
that vanish at infinity.
The aim of the present section is to study some aspects of GL (G) as a C∗-subalgebra of
0
GL(G). Let us give a simple but important result whose proof involves nothing more than
routine calculations.
Lemma 3.2 Suppose that K(G) is directed downwards and for each α, u ∈ C (G) is chosen
Kα c
such that 0 ≤ u ≤ 1 and u (x) = 1 for all x ∈ K . Then (u ) is a bounded approximate
Kα Kα α Kα
identity for GL (G).
0
Our next result shows that the subspaces
M(G)◦GL (G) := ζ ◦f : ζ ∈ M(G) and f ∈ GL (G)
0 0
n o
and
GL (G)◦M(G) := f ◦ζ : ζ ∈ M(G) and f ∈ GL (G)
0 0
n o
of GL(G) coincide with GL (G).
0
Lemma 3.3 The following assertions hold.
(i) M(G)◦GL (G) = GL (G),
0 0
(ii) GL (G)◦M(G) = GL (G).
0 0
Second dual of measure algebra 5
Proof. We prove the first; the proof of the second is similar. Since the inclusion GL (G) ⊆
0
δ ◦GL (G) ⊆ M(G)◦GL (G), it will be enough to prove the reverse inclusion. To this end, let
e 0 0
ζ ∈ M(G), f = (f ) ∈ GL (G) and ǫ > 0 be given. Without loss of generality, we may
µ µ∈M(G) 0
assume that ζ is non-zero and positive and that f 6= 0. By the regularity of ζ, we can choose a
compact subset K of G such that 0 < ζ(G \K ) < ǫ/2kfk . Also, since f vanishes at infinity,
1 1 ∞
there is a compact subset K in G with kf −χ fk < ǫ/2kζk. Therefore
2 K2 ∞
kζ ◦f −(χ ζ)◦(χ f)k ≤ kζ −χ ζkkfk +kχ ζkkf −χ fk ≤ ǫ, (1)
K1 K2 ∞ K1 ∞ K1 K2 ∞
where χ ζ is the measure in M(G) defined on each Borel subsets A of G by χ ζ(A) =
K1 K1
χ dζ. Now suppose that µ is an arbitrary element of M(G). Observe that
A K1
R
(χ ζ)◦(χ f) (x) = χ (yx)f (yx)dζ(y).
(cid:16) K1 K2 (cid:17)µ ZK1 K2 (χK1ζ)∗µ
For each x ∈ G \ K −1K , we get K x ⊆ G \ K , and hence (χ ζ) ◦ (χ f) (x) = 0 for
1 2 1 2 (cid:16) K1 K2 (cid:17)
µ–almost all x ∈ G \K −1K . Thus, inequality (1) implies that
1 2
|(ζ ◦f) (x)| < ε, for µ−almost all x ∈ G \K −1K .
µ 1 2
It follows that ζ ◦f ∈ GL (G). We have now completed the proof of the lemma. (cid:4)
0
Now, let M (G) be the closed ideal of M(G) consisting of all absolutely continuous measures
a
with respect to λ and let L1(G) denote the group algebra of G as defined in [9, Theorem 14.17
and14.18]. Then,theRadon-NikodymTheoremcanbeinterpretedasanidentification ofM (G)
a
with {ν : ϕ ∈ L1(G)}, where ν is the measurein M(G) defined on each Borel subsetA of G by
ϕ ϕ
ν (A) = ϕdλ. This allows us to show that M (G)∗, the first dual space of M (G), is L∞(G),
ϕ A a a
where L∞R(G) denotes the Lebesgue space as defined in [9, Definition 12.11] equipped with the
essential supremum norm. Given any σ ∈ M (G) and g ∈ L∞(G), define the complex-valued
a
functions g⋆σ and σ⋆g on G by
(g⋆σ)(x) = hg,δ ∗σi = g(xy) dσ(y)
x Z
G
and
(σ⋆g)(x) = hg,σ∗δ i = g(yx) dσ(y)
x Z
G
for all x ∈ G, where δ denotes the Dirac measure at x. Then, it is not hard to check that
x
the functions g ⋆ σ and σ ⋆ g are in C (G); this is because of, M (G) can be identified with
b a
all ν ∈ M(G) such that the map x 7→ δ ∗|ν| and x 7→ |ν|∗ δ from G into M(G) are norm
x x
continuous, see for example [9, 19.27 and 20.31 ]. In particular, if P : GL(G) → L∞(G) is the
adjoint of the natural embedding from M (G) into M(G), then P is the restriction mapping and
a
hence norm decreasing and onto.
For the formulation of the following statements we recall Remark 2.2 which allows us to
consider C (G) as a closed subspace of GL(G) containing C (G).
b 0
6 H. Javanshiri and R. Nasr-Isfahani
Lemma 3.4 If σ is an arbitrary element of M (G), then for given f = (f ) in GL(G)
a µ µ∈M(G)
and all µ ∈ M(G), we have
(i) (σ◦f) = σ⋆P(f), |µ|–a.e.,
µ
(ii) (f ◦σ) = P(f)⋆σ, |µ|–a.e.
µ
In particular, (σ◦f) and (f ◦σ) are in C (G) for all µ ∈ M(G).
µ µ b
Proof. We prove the assertion (i); the proof of (ii) is similar. First note that σ ◦ f is the
generalised function h = (h ) , where for each µ ∈ M(G) satisfies the following equality
µ µ∈M(G)
h (x) = (σ◦f) (x) = l f (x) = f (yx) dσ(y).
µ µ σ σ∗µ Z σ∗µ
G
On the other hand, for an arbitrary µ in M(G) and any Borel subset A of G, since χ µ ≪ µ,
A
we have
χ h dµ= f d(σ ∗χ µ)
Z A µ Z σ∗χAµ A
G G
=hP(f),σ ∗χ µi
A
= P(f)(yx) dσ(y)d(χ µ)(x)
Z Z A
G G
= χ (σ⋆P(f)) dµ.
Z A
G
Hence, (σ◦f) = σ⋆P(f) |µ|–a.e. (µ ∈ M(G)). It follows that σ◦f ∈ C (G). (cid:4)
µ b
Now, in light of Lemmas 3.3 and 3.4, the following proposition is now immediate.
Proposition 3.5 The following assertions hold.
(i) M (G)◦GL (G) = C (G),
a 0 0
(ii) GL (G)◦M (G) = C (G).
0 a 0
Recall from [15, Page. 90] that a linear functional m in GL(G)∗ is called a mean if m(1) = 1
and m(f)≥ 0 whenever f ∈ GL(G) with f ≥ 0, and it is topological left invariant if m(ζ◦f)=
m(f) for all f ∈ GL(G) and
ζ ∈ P(G) = ν ∈ M(G) : ν ≥ 0 and kνk = 1 .
n o
In [15, Theorem 4.1], Wong proved that GL(G) has a topological left invariant mean if and only
if M(G)∗ has a topological left invariant mean. In particular, he showed that Ψ∗, the adjoint of
Ψ, maps the set of all topological left invariant means on M(G)∗ onto that of GL(G). Related to
this result, we have the following result which asserts that in the case where G is non-compact,
thenanytopological leftinvariantmeanonGL(G)liesinGL (G)⊥,wherehereandinthesequel,
0
GL (G)⊥ denotes the following weak∗-closed subspace of GL(G)∗
0
m∈ GL(G)∗ : hm,fi= 0 for all f ∈ GL (G) .
0
n o
In fact, the next result shows that GL (G)⊥ is far from devoid of interest.
0
Second dual of measure algebra 7
Proposition 3.6 If G is non-compact, then any topological left invariant mean on GL(G) lies
in GL (G)⊥.
0
Proof. Suppose that m is a topological left invariant mean on GL(G). First note that the non-
compactnessofG impliesthatthereexistsasequence(x )ofdisjointelementsofG andacompact
n
symmetric neighborhood V of e such that the sets x V for all n ∈ N are pairwise disjoint; see
n
[9, 11.43(e)]. Now, it is not hard to check that χ = δ ◦χ , |µ|−a.e. for all x ∈ G and
xV x−1 V
µ ∈ M(G). Moreover, by Remark 2.2, for each p ∈ N the function p χ = p δ ◦χ
n=1 xnV n=1 xn−1 V
is in GL(G) for which p χ ≤ 1. It follows that P P
n=1 xnV
P
p
phm,χ i = hm, χ i ≤ 1 (p ∈ N).
V xnV
X
n=1
Thus hm,χ i = 0 and thus we have
V
hm,χ i = hm,δ ◦χ i = 0 (x ∈ G). (2)
xV x−1 V
Now suppose that f = (f ) is a non-zero element of GL (G). The proof will be
µ µ∈M(G) 0
completed by showing that hm,fi = 0. To this end, without loss of generality, we may assume
that kfk = 1. Then, since f∗f = (|f |2) vanishes at infinity, for given ε > 0 one can
∞ µ µ∈M(G)
choose y ,...,y ∈ G such that
1 q
q
|f |2 ≤ χ +ε |µ|−a.e. µ ∈ M(G) .
µ yiV
X (cid:16) (cid:17)
i=1
Now, by considering h = q χ +ε ∈ BM(G) as an element of GL(G), we have f∗f ≤ h,
i=1 yiV
see Remark 2.1. Hence, inPlight of [13, Theorem 3.3.2] and equality (2), we see that
|hm,fi|2 ≤ hm,f∗fi≤ hm,hi ≤ ε.
It follows that hm,fi= 0. Hence, m ∈ GL (G)⊥. (cid:4)
0
4 GL (G)∗ as a subalgebra of M(G)∗∗
0
As we know, there exists two natural products on M(G)∗∗ extending the one on M(G), known
as the first and second Arens productsof M(G)∗∗. Thefirst Arens producton M(G)∗∗ is defined
in three steps as follows. For m, n in M(G)∗∗, the element m⊙n of M(G)∗∗ is defined by
hm⊙n,fi = hm,nfi (f ∈ GL(G)),
wherehmf,ζi= hm,fζiandfζ = ζ◦f forallζ ∈ M(G). Equippedwiththisproduct,M(G)∗∗ isa
BanachalgebrawhichcontainsM(G)asasubalgebra. Moreover, bythedualityrelation between
M(G) and GL(G), there exists a unique generalised function h ∈ GL(G) such that mf = Ψ(h);
In what follows, we denote the generalised function Ψ−1(mf) corresponding to mf ∈ M(G)∗, by
8 H. Javanshiri and R. Nasr-Isfahani
mf. Moreover, M(G) and GL (G) are in duality with respect to the natural bilinear map given
0
for each η ∈ M(G) and f = (f ) in GL (G) by hf,ηi = f dη. Therefore, M(G) may
µ µ∈M(G) 0 G η
be identified with a closed subspace of GL (G)∗. Furthermore, ifRf = (f ) ∈GL (G) and
0 µ µ∈M(G) 0
ζ ∈ M(G), then, by Lemma 3.3, ζ ◦f and f ◦ζ are also in GL (G) and
0
hζ ◦f,νi= hf,ζ ∗νi, and hf ◦ζ,νi= hf,ν ∗ζi,
for all ν ∈ M(G). Hence the product “⊙” is well defined on GL (G)∗ and GL (G)∗ is a Banach
0 0
algebra with this product, if we show that GL (G) is a topologically introverted subspace of
0
GL(G). To this end, we have the following result.
Proposition 4.1 The space GL (G) is left (right) topologically introverted in GL(G); that is,
0
mf ∈ GL (G) (fm∈ GL (G)) for all m ∈ GL (G)∗ and f ∈GL (G).
0 0 0 0
Proof. We only show that GL (G) is a left topologically introverted subspace of GL(G); the
0
proof of the other assertion is similar. To this end, let m∈ GL (G)∗, f = (f ) ∈ GL (G)
0 µ µ∈M(G) 0
and ε > 0 be given. Since GL (G) is spanned by its positive elements, we can suppose that
0
m ≥ 0. Also, since f vanish at infinity, there is a compact set B in G with |f (x)| < ε for
µ
µ-almost all x ∈ G \B (µ ∈ M(G)).
Now let ̺ denote the restriction of m to C (G). Then there exists a compact subset K of
0
G such that ̺(G \K) < ε/2. In particular, if m denote the continuous linear functional on
K
GL (G) defined by
0
hm ,hi := hm,h−u hi h ∈ GL (G) ,
K K 0
(cid:16) (cid:17)
where u is a fixed function in C (G) such that 0 ≤ u ≤ 1, and u (x) = 1 for all x ∈ K.
K c K K
Thenthepositivity ofthelinear functionalm onGL (G)impliesthatkm k = lim hm ,u i,
K 0 K α K Kα
where (u ) is the net introduced in Lemma 3.2. Hence, there exists α such that
Kα 0
ε
km k− ≤ hm ,u i≤ km | k.
K 2 K Kα0 K C0(G)
It follows that km k < ε. Indeed,
K
km | k=sup |h̺,g−u gi| : g ∈ C (G) and kgk ≤ 1
K C0(G) n K 0 o
=sup hχ ̺,(g−u g)i : g ∈ C (G) and kgk ≤ 1
G\K K 0
n(cid:12) (cid:12) o
(cid:12) (cid:12)
≤kχG\K(cid:12)̺k (cid:12)
=̺(G \K).
If now, ν is an arbitrary probability measure in M(G), then ζ := (χ )ν is a measure
G\BK−1
in M(G) for which supp(ζ) ⊆ G \BK−1. Further, choose a compact subset D in G for which
D ⊆ G\BK−1 andthat|ζ|(G\D) < ε. Trivially, foreach x ∈ G\D−1B,wesee thatDx ⊆ G\B,
and therefore for each µ ∈ M(G) we have
|(ζ ◦f) (x)| ≤ |f (yx)| d|ζ|(y)+ |f (yx)| d|ζ|(y) ≤ ε(kfk +1);
µ Z ζ∗µ Z ζ∗µ ∞
G\D D
Second dual of measure algebra 9
that is, |(ζ ◦ f) (x)| ≤ ε(kfk + 1) for µ-almost all x ∈ G \ D−1B. In particular, since
µ ∞
D−1B ∩K = ∅, we see that
k(ζ ◦f)χ k = sup k(ζ ◦f) χ k ≤ ε(kfk +1).
K ∞ µ K µ,∞ ∞
µ∈M(G)
Thus
(mf) (x) dζ(x)=hmf,ζi
ZG\BK−1 ζ
=hm,u (ζ ◦f)i+hm ,ζ ◦fi
K K
≤ε(kfk +1)kmk+εkζkkfk .
∞ ∞
On the other hand, since ζ ≪ ν, we have
(mf) (x) dζ(x)= (mf) (x) dζ(x) = (mf) (x) dν(x).
ZG\BK−1 ζ ZG\BK−1 ν ZG\BK−1 ν
This shows that, if ν ∈ M(G), then (mf) (x) ≤ ε[(kfk + 1)kmk + kfk ] for ν-almost all
ν ∞ ∞
x ∈ G \BK−1, and thus mf ∈ GL (G). (cid:4)
0
A linear functional m in GL (G)∗ (resp. M(G)∗∗) has compact carrier if there exists a
0
compact set K in G such that hm,fi = hm,χ fi for all f ∈ GL (G) (resp. GL(G)); such a
K 0
compact set K is called a compact carrier for m. In the sequel, the notation M (G)∗∗ is used to
c
denote the norm closure of functionals in M(G)∗∗ with compact carrier.
Now, with an argument similar to the proof of Propositions 2.6, 2.7 and Theorems 2.8 and
2.11 in [11], one can prove the following result which in particular shows that the restriction
map is an isometric algebra isomorphism fromM (G)∗∗ onto GL (G)∗. In other word, this result
c 0
allows us to view GL (G)∗ as a subalgebra of M(G)∗∗.
0
Theorem 4.2 The following assertions hold.
(i) Functionals in GL (G)∗ with compact carriers are norm dense in GL (G)∗.
0 0
(ii) If m and n are elements in GL (G)∗ (resp. M (G)∗∗) with compact carriers K and K′
0 c
respectively, then m⊙n has compact carrier KK′.
(iii) The restriction map is an isometry and an algebra isomorphism from M (G)∗∗ onto
c
GL (G)∗.
0
(iv) M(G)∗∗ = GL (G)∗⊕GL (G)⊥. In fact, any m ∈ M(G)∗∗ has a unique decomposition
0 0
m = m +m , where m ∈ GL (G)∗, m ∈ GL (G)⊥ and kmk= km k+km k. Moreover,
∗ ⊥ ∗ 0 ⊥ 0 ∗ ⊥
m ≥ 0 if and only if m ≥ 0 and m ≥ 0.
∗ ⊥
(v) GL (G)⊥ is a weak∗-closed ideal of M(G)∗∗.
0
(vi) GL (G)∗ is a left or right ideal of M(G)∗∗ if and only if G is compact.
0
10 H. Javanshiri and R. Nasr-Isfahani
(vii) M(G) is a left or right ideal of GL (G)∗ if and only if G is discrete.
0
(viii) M (G) is a two-sided ideal in GL (G)∗.
a 0
Proof. The details are omitted and we only give the proof for (ii) and (viii).
(ii) Suppose that m and n are elements in GL (G)∗ (resp. M (G)∗∗) with compact carriers
0 c
K and K′, respectively, and that f is an arbitrary element in GL (G) (resp. GL(G)). First,
0
observe that
hm⊙n,fi= hm,nfi= hm,χ (nf)i.
K
On the one hand, for µ ∈ M(G) we have ζ := χ µ ≪ µ, and so
K
hχK(nf),µi = Z χK(nf)µ dµ = Z (nf)ζ dζ = hn,χK′(ζ ◦f)i.
G G
On the other hand, χK′(ζ ◦f) = ζ ◦(χKK′f). Indeed, for each ν ∈ M(G) and h ∈ C0(G), we
have
(cid:16)χKK′(ζ ∗ν)(cid:17)(h)=ZGχKK′hd(ζ ∗ν)
=Z Z χKK′(xy)χK(x)h(xy) dµ(x) dν(y)
G G
=Z Z χK′(y)χK(x)h(xy) dµ(x) dν(y)
G G
= ζ ∗(χK′ν) (h),
(cid:16) (cid:17)
and this implies that
hχK′(ζ ◦f),νi = hζ ◦f,χK′νi= hf,χKK′(ζ ∗ν)i = hζ ◦(χKK′f),νi.
Hence by using these equalities, we have
hχK(nf),µi = hn,ζ ◦(χKK′f)i= hn(χKK′f),ζi= hχK(n(χKK′f)),µi.
Consequently
hm⊙n,fi= hm,χK(nf)i= hm,χK(n(χKK′f))i= hm,n(χKK′f)i= hm⊙n,χKK′fi.
It follows that m⊙n has compact carrier KK′.
(viii) ThatM (G) is a closed subalgebraof GL (G)∗ is trivial. Now, supposethat σ ∈ M (G)
a 0 a
and m ∈ GL (G)∗. We show that m⊙σ ∈ GL (G)∗; that σ ⊙m ∈ GL (G)∗ is similar. Let ζ
0 0 0
denote the restriction of m to C (G). Since M (G) is an ideal in M(G), we have ζ ∗σ ∈ M (G).
0 a a
We now invoke Proposition 3.5 to conclude that
hm⊙σ,fi = hζ,f ◦σi = hζ ∗σ,fi f ∈ GL (G) ,
0
(cid:16) (cid:17)
whence m⊙σ = ζ ∗σ ∈ M (G). (cid:4)
a