Iranian Journal of Science & Technology, Transaction A, Vol. 28, No. A2
Printed in Islamic Republic of Iran, 2004
© Shiraz University
REAL GROUP ALGEBRAS
*
A. EBADIAN 1, 2 AND A. R. MEDGHALCHI 1 **
1
Faculty of Mathematics, Teacher Training University, Tehran, I. R. of Iran, 15614
2
Current address: Department of Mathematics, Urmia University, I. R. of Iran
a.ebadian@mail.urmia.ac.ir, medghalchi@saba.tmu.ac.ir
Abstract – In this paper we initiate the study of real group algebras and investigate some of its aspects.
1
Let L (G ) be a group algebra of a locally compact group G, τ : G → G be a group homeomorphism
2
p
p
such that τ = τοτ = 1 , the identity map, and L (G , τ ) = { f ∈ L (G ) : fοτ = f } ( p ≥ 1) . In this
p
paper, among other results, we clarify the structure of L (G , τ ) and characterize amenability of
L1 (G, τ ) and identify its multipliers.
Keywords – Real Banach algebra, amenability, multiplier, derivation, group involution
1. INTRODUCTION
In 1965, Ingelstam [1] introduced the theory of real Banach algebras. The real function algebra theory
was developed further by Kulkarni and Limaye [2]. In their excellent monograph, “Real function
algebras”, Kulkarni and Limaye present interesting aspects of the theory of C ( X ,τ ) . We refer to [3]
for our notations.
Let G be a locally compact group. An automorphism τ : G → G is called a topological group
involution on G if τ is a homeomorphism and τ (τ ( x)) = x for all x ∈ G . For example, in group
(C ,+)τ ( z ) = z and in ( R \ {0},.),τ ( x) = x −1 are topological group involutions. Note that we do not
assume that τ ( xy ) = τ ( y )τ ( x) .
Let Co (G, τ ) = { f ∈ Co (G) : foτ ( x) = f ( x ), x ∈ G} , and C c (G, τ ) = { f ∈ C c (G ) :
f o τ ( x) = f ( x) , x ∈ G} it is clear that, if τ is the identity map on G , then
C o (G, τ ) = C or (G ) , C c (G , τ ) = C cr (G ) . If 1 ≤ p ≤ ∞ , we define f o τ (x)= f (x) , for all x ∈ G} .
Clearly, L p (G, τ ) ⊆ L p (G ) and if τ is the identity map, L p (G,τ ) consists of real functions.
2. THE STRUCTURE OF L1 (G,τ ) AND M (G, τ )
Lemma 2. 1. Let G be a locally compact group and τ be a topological group involution on G . If
σ : C c (G ) → C c (G ) is defined by σ : ( f ) = f oτ , then (i) σ is an algebra involution on
C c (G ) and C c (G, τ ) = { f ∈ C c (G ) | σ ( f ) = f },
(ii) C c (G ) = C c (G ,τ ) ⊕ iC c (G, τ ) .
∗
Received by the editor May 14, 2002 and in final revised form January 26, 2004
Corresponding author
∗∗
290
A. Ebadian / A. R. Medghalchi
Proof. (i) We must show that whenever f ∈ C c (G ) , then fo τ ∈ C c (G ) . To do this, we have
supp ( foτ ) = cl ([( foτ ) −1{0}]′ ⊆ τ −1 (supp f ) .
It follows that supp ( foτ ) is compact, i.e., ( fo τ ) ∈ C c (G ) . Hence, supp ( foτ ) is compact, i.e.,
foτ ∈ C c (G ) . The rest of (i) is clear.
( f − σ ( f )) . Since σ 2 = i , ( = identity)
f +σ(f )
( f + σ ( f ))
(ii) Clearly,
f =
+i
σ(
2
2i
2
f −σ( f )
f − σ ( f ) . It follows that f = g + ih where g , h ∈ C (G , τ ) .
f + σ ( f ) and
)=
σ(
c
2i
2i
2
Now if f = g + ih = g1 + ih1 , then g = f + σ ( f ) ,i.e., g = g 1 and thus h = h1 .
2
)=
Note. By the same argument one can conclude that C 0 (G ) = C 0 (G ,τ ) ⊕ iC 0 (G, τ ) . In fact it is
enough to show that foτ ∈ C 0 (G ) whenever f ∈ C 0 (G ) . Since f ∈ C 0 (G ) , for a given ε > 0 ,
there is a compact set F in G such that | f ( x) |< ε whenever x ∈ F ′ . Clearly, τ −1 ( F ) is compact,
and if x ∉ τ −1 ( F ) , then τ ( x) ∉ F ,i.e., | foτ ( x ) |< ε . Therefore, foτ ∈ C 0 (G ) .
Let M (G ) be the Banach space of all complex regular Borel measures on G . For each
µ ∈ M (G ) , we define µτ = µoτ , then it is clear that µτ ∈ M (G ) . Also by Lebesgue dominated
convergence theorem one can show that for every bounded Borel measurable function h on G ,
∫
G
hdµτ = ∫ (hoτ )dµ .
G
(1)
Clearly, (1) is true when h is a characteristic function; by linearity it holds when h is a simple
function; by continuity (1) holds when h is integrable.
Proposition 2. 2. Let M (G,τ ) = {µ ∈ M (G ) | µoτ = µ } . Then M (G, τ ) is a real Banach algebra
−1
with the convolution product µ ∗ν (E) = ν ( x E)dµ ( x) = µ ( Ey −1 )dν ( y ) ( µ ,ν ∈ M (G, τ ))
G
G
and M (G ) = M (G, τ ) ⊕ iM (G, τ ).
∫
∫
Proof. Let µ ,ν ∈ M (G ,τ ). Then
( µ ∗ν )oτ ( E ) = ∫ ν ( x −1τ ( E ))dµ ( x) = ∫ ν (τ (τ ( x) −1 E ))dµ ( x) .
G
G
= ∫ ν ((τ ( x) −1 ) E )dµ ( x) = ∫ ν ( x −1 E )dµoτ
G
G
(2)
= µ ∗ν (E )
Therefore µ ∗ν ∈ M (G, τ ) . The rest of the proof follows the same line as the proof of Lemma 2.1.
Therefore, it is omitted.
Remark. For a real linear space A , the real dual space of A , that is, the space of all real-valued
continuous linear functional on A will be denoted by A∗ .
Proposition 2. 3. Every real-valued continuous functional φ on C 0 (G, τ ) can be represented as
φ ( f ) = fdµ , where µ is the unique measure in M (G , τ ) such that || ψ ||=|| µ || and vice versa.
∫
G
Then f = g + ih where g , h ∈ C 0 (G, τ ) . If we define
ψ ( f ) = φ ( g ) + iφ (h) , then clearly ψ ∈ C 0 (G ) ∗ and so by the Riesz representation theorem ([3,
Proof.
Let
f ∈ C 0 (G, τ ) .
Iranian Journal of Science & Technology, Trans. A Volume 28, Number A2
Summer 2004
291
Real group algebras
exists a unique measure µ
in M (G ) such that
ψ ( f ) = fdµ ( f ∈ C 0 (G )) and || ψ ||=|| µ || . It follows that φ (h) = hdµ for every h in
G
G
C 0 (G , τ ) .
Now,
in
order
to
prove
that
µ ∈ M (G,τ ) ,
we
have
ψ (σ ( f )) = ψ ( g − ih) = φ ( g ) − iφ (h) = ψ ( f ) . Therefore,
theorem
∫
(14.4)]),
there
∫
G
∫
fdψ = ∫ σ ( f )dµ = ∫ σ ( f )d µ = ∫ fdµ oτ
G
G
G
(3)
( f ∈ C 0 (G )) . Thus, µ = µoτ , i.e. µ ∈ M (G,τ ) . Also, similar to the proof of [6, Theorem 3.2.1] we
can show that || ψ ||=|| µ || .
Conversely, let µ ∈ M (G,τ ) and φ ( f ) = ∫ fd µ ( f ∈ C 0 (G ,τ )) . If f ∈ C0 (G,τ ) , then
G
σ ( f ) = f . Hence,
φ ( f ) = φ (σ ( f )) = ∫ σ ( f )dµ = ∫ σ ( f )dµ = ∫ ( foτ )dµ
G
=
∫
G
G
fd µ o τ =
∫
G
G
(4)
fd µ = φ ( f ) .
Thus φ ( f ) is real.
Theorem 2. 4. Let G be a locally compact group with the left Haar measure λ and τ be a
topological group involution on G . Then λ o τ = λ .
Proof. It is easy to show that λ o τ is a positive measure on G . Also if B is a Borel set, then
λoτ ( xB ) = λ (τ ( xB )) = λ (τ ( x )τ ( B )) = λ (τ ( B )) = λoτ ( B )( x ∈ G ) . Therefore, λ o τ is left
invariant. So, there is a positive number c such that λoτ ( B ) = cλ ( B ) for every Borel set B . If U
is an open set, then λ o τ (τ (U )) = c λ (τ (U )) ,i.e., λ (U ) = c λ (τ (U )) which is equal to
c 2 λ (U ) . Therefore, for every open set U we have λ (U ) = c 2 λ (U ) . So, c = 1 . Hence,
λoτ = λ .
For a locally compact group G and the Haar measure λ we defined
p
L (G, τ ) = { f ∈ L p (G ) | foτ = f }(1 ≤ p ≤ ∞). Clearly L p (G ,τ ) ⊆ L p (G ), L p (G, τ ) is a real
algebra and L p (G ) = L p (G, τ ) ⊕ iL p (G, τ ).
Theorem 2. 5. (a) For 1 ≤ p ≤ ∞ , L p (G, τ ) is a real Banach space, and L2 (G, τ ) is a real Hilbert
space with an inner product,
< f , g >= ∫ gdλ .
G
(5)
(b) For each f , g ∈ L1 (G ,τ ), max{|| f || p , || g || p } ≤|| f + ig ||≤|| f || p + || g || p .
(c) L1 (G, τ ) ∗ = L∞ (G, τ ) .
(d) L1 (G,τ ) has a bounded approximate identity of norm 1.
Proof. (a). Clearly, L p (G, τ ) is a real subspace of L p (G ) . Let f , g ∈ L p (G, τ ) then
f ∗ g ∈ L p (G ), [4]. We will show that f ∗ g ∈ L p (G, τ ) . In order to do this, by (2.4) and (1) we
have
( f ∗ g )(τ ( x )) =
Summer 2004
∫
G
f ( y ) g ( y −1τ ( x )) d λ ( y )
Iranian Journal of Science & Technology, Trans. A, Volume 28, Number A2
292
A. Ebadian / A. R. Medghalchi
=
∫
G
f (τ (τ ( y ))) g (τ (τ ( y )) −1 x )) dλ ( y )
=
∫
G
=
=
∫
G
∫
(6)
f (τ ( y )) g (τ ( y −1 ) x )d λ ( y )
G
f ( y ) g ( y −1 x )d λ o τ ( y )
f ( y ) g ( y −1 x )d λ ( y ) = ( f ∗ g ) ( x )
for every x ∈ G , hence f ∗ g ∈ L p (G, τ ) . We now prove that L p (G, τ ) is complete. Let { f n }∞n =1 be
a Cauchy sequence in L p (G, τ ) . Since L p (G ) is complete, there exists f ∈Lp (G) such that
lim n→∞ || f n − f || p = 0 . Now, there exists a subsequence of { f n }∞n =1 as { f n k }∞k =1 such that
λ -almost everywhere, and so f (τ ( x)) = lim k →∞ f n (τ ( x)) = lim k →∞ f n ( x) = f ( x), λ almost everywhere. Therefore, f ∈ L p (G, τ ) . Hence L p (G, τ ) is a real Banach algebra and not a
lim k →∞ f nk ( x ) = f ( x ) ,
k
k
complex algebra.
If < f , g >=
∫
G
fgdλ for every
f , g ∈ L2 (G, τ ) , then < f , g >= < f , g > . Therefore
L2 (G,τ ) is a real Hilbert space.
1
(b) For f , g ∈ L1 (G, τ ) we have || f || p ≤ (|| ( f + ig ) || p + || ( f − ig ) || p ) =|| f + ig || p .
2
Similarly, || g || p ≤|| f + ig || p .
1
*
∞
(c) We know that L (G ) ≅ L (G ) . Let f ∈ L1 (G ) . So f = g + ih where g, h ∈ L1 (G,τ ) .
Now, we define ψ ( f ) = φ ( f ) + iφ ( g ) where φ ∈ L1 (G, τ ) * . It is clear that ψ ∈ L1 (G ) * and
therefore, there exists a unique p ∈ L∞ (G ) such that ψ ( f ) = ∫ f pdλ ( f ∈ L1 (G )) .
G
Hence we have,
ψ (σ ( f )) = ψ ( g − ih) = φ ( g ) − iφ (h) = ψ ( f ) .
(7) (∗)
This implies that
∫
G
f pdλ = ∫ σ ( f ) pdλ = ∫ ( foτ ) pdλ = ∫ f poτdλ ( f ∈ L1 (G )) .
G
G
(8)
G
∫
Therefore, poτ = p , i.e., p ∈ L∞ (G, τ ) . Also, we have φ ( f ) =
f pdλ for every
G
1
1
f ∈ L (G, τ ) and by (∗) φ ( f ) is real. Conversely, if φ : L (G, τ ) → R is defined by φ ( f ) = f
G
where p ∈ L∞ (G, τ ) and f is an arbitrary function, then φ ∈ L1 (G, τ ) * and the proof is complete.
∫
(d) Let U be any compact neighborhood of e and (U α ) be the collection of all compact
neighborhoods of e in U , which is directed by a set inclusion (α ≤ β if and only if U α ⊇ U β ) . If
and g α = χ τ (U ) = χ U α oτ , then, since τ is a homeomorphism, { f α } and
λ (U α ) λ (τ (U α ))
λ (U α )
{g α } are bounded approximate identities of norm one for L1 (G ) . If we define eα = f a + g α , then
we define f α = χ U α
α
2
{eα } is a bounded approximate identity of norm one for L1 (G ) , and also for L1 (G, τ ) since
eα ∈ L1 (G, τ ) .
Lemma 2. 6. For 1 ≤ p ≤ ∞ , the linear space C c (G, τ ) is a dense subspace of L p (G,τ ) .
Iranian Journal of Science & Technology, Trans. A Volume 28, Number A2
Summer 2004
Real group algebras
293
Proof. Suppose that f ∈ L p (G, τ ) , since C c (G ) is a dense subspace of L p (G ) , there exists a
f + f n oτ
.Then
sequence { f n }∞n =1 in C c (G ) such that lim n →∞ || f n − f || p = 0 . Let g n = n
2
g n ∈ C c (G, τ ) and limn→∞ || gn − ( f + foτ ) || p = limn→∞ || gn − f || p = 0 .
2
Theorem 2. 7. For µ ∈ M (G, τ ) and ψ ∈ L2 (G, τ ) , let Tµψ = µ ∗ψ . Each Tµ is a bounded
operator on the real Hilbert space L2 (G, τ ) , and the mapping µ → Tµ is a faithful
∗ − representation of M (G,τ ) . Note that M (G,τ ) is a ∗ − Banach algebra.
Proof. The linearity of Tµ on L2 (G, τ ) is obvious, and the boundedness of Tµ , with || Tµ ||≤|| µ || ,
follows from [3,(20.12.ii)]. For ψ ∈ L1 (G, τ ) ∩ L2 (G, τ ) , we have
( µ ∗ν ) ∗ψ = µ ∗ (ν ∗ψ )
(9)
[2, (19.2.iv)]. Thus Tµ ∗ν (ψ ) = Tµ (Tνψ ) for all ψ ∈ L1 (G, τ ) ∩ L2 (G, τ ) . Since C c (G , τ ) ⊆ L1 (G , τ ) ∩ L2 (G , τ ) ,
by Lemma (2.7), L1 (G, τ ) ∩ L2 (G, τ ) is dense in L2 (G, τ ) . It follows that Tµ ∗ν = Tµ Tν . To show that
f ∗ dµ ≠ 0 . Since µ ∗ f (e ) = ∫ f ∗ dµ ≠ 0
Tµ ≠ 0 if µ ≠ 0 , consider an f ∈ C c (G , τ ) such that
G
G
and µ ∗ f is continuous; thus Tµ f is not a zero element of L2 (G, τ ) . Note that f ∗ is the involution
of f .
∫
3. AMENABILITY AND WEAK AMENABILITY OF REAL GROUP ALGEBRAS
In this section, we show that amenability of L1 (G,τ ) and L1 (G ) are equivalent. We shall use some
notions of [1].
Definition 3. 1. A Banach algebra A over F is called amenable if for every Banach A-module X
over F , H 1 ( A, X ∗ ) = {0} .
Let A be a Banach algebra over F , and X be a Banach A-module over F . If F = R , we say
that X is a real Banach A-module for the real Banach algebra A . If F = C , we say X is a Banach
A-module for the Banach algebra A .
Definition 3. 2. Let X be a real Banach space. Then BLR ( X , C ) , consists of all complex-valued
continuous real-linear functional on X , which is a real Banach space, denoted by X ′ and called the
complex dual of X .
If A is a real Banach algebra and X is a real Banach A -module, then X ′ with the natural module
action is also a real Banach A -module.
Note that in this case X ′ is isomorphic to X ∗ × X ∗ .
Lemma 3. 3. Let G be a locally compact group and let τ be a topological involution on G .
Suppose X is a real Banach L1 (G,τ ) -module. Then H 1 ( L1 (G, τ ), X ′) = {0} if and only if
H 1 ( L1 (G, τ ), X ∗ ) = {0} .
Proof. It is easy to see that Z 1 ( L1 (G, τ ), X ′) = Z 1 ( L1 (G, τ ), X ∗ ) ⊕ iZ 1 ( L1 (G, τ ), X ∗ ) . Now, let
H 1 ( L1 (G, τ ), X ∗ ) = {0} and let D ∈ Z 1 ( L1 (G, τ ), X ′). There exist elements a and b in X ∗ such
that D = δ a + iδ b . If c = a + ib , then c ∈ X ′ and d = δ c . Hence H 1 ( L1 (G, τ ), X ′) = {0} .
Summer 2004
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A. Ebadian / A. R. Medghalchi
Conversely, we assume that H 1 ( L1 (G, τ ), X ′) = {0} and let D ∈ Z 1 ( L1 (G, τ ), X ∗ ) . By the
assumption D ∈ B 1 ( L1 (G , τ ), X ′) . Clearly, B 1 ( L1 ( G , τ ), X ′) = . B1 ( L1 (G,τ ), X ∗ ) ⊕ iB1 ( L1 (G,τ ), X ∗ )
Hence there exist unique elements D1 , D2 in B 1 ( L1 (G, τ ), X ∗ ) such that D = D1 + iD2 . On the
other hand, D = D + i 0 where D,0 ∈ Z 1 ( L1 (G , Z ), X ∗ ) . Therefore, we have D1 = D and
D2 = 0 . Hence D ∈ B 1 ( L1 (G , τ ), X ∗ ) and so H 1 ( L1 (G, τ ), X ∗ ) = {0} .
Lemma 3. 4. Let ( X , || . ||) be a real Banach space and X × X be the (complex) linear space under
the standard operations of addition and scalar multiplication. If we equip X × X by the norm || | . || | ,
which satisfies the inequalities
max{|| x ||, || y ||} ≤ C1 || |
(10)
and
|| | ( x, y ) || |≤ C 2 max{|| x ||, || y ||},
(11)
for constants C1 and C 2 , then
(i) ( X × X , ||| . |||) is a Banach space
(ii) The map η : X → X × X , defined by η ( x) = ( x,0) , is a real-linear continuous mapping.
(iii) The map ψ : X ′ → ( X × X ) ∗ , defined by ψ (λ )( x, y ) = λ ( x) + iλ ( y ) , is a real-linear
continuous mapping onto the real Banach space ( X × X ) ∗ .
Proof. (i) and (ii) are clear. (iii) ψ is a well-defined real-linear mapping. For each λ ∈ X ′ we have
|| ψ (λ ) ||= sup{| ψ ( x, y ) |:|| | ( x, y ) || |≤ 1, x, y ∈ X }
≤ sup{| λ ( x) | + | λ ( y ) |:|| x ||≤ C1 , || y ||≤ C1 , x, y ∈ X }
(12)
≤ 2C1 || λ || .
Hence ψ is continuous. On the other hand, for each λ ∈ X ′ we have
|| ψ (λ ) ||= sup{| ψ (λ )( x,0) |: x ∈ X , || | ( x,0) || |≤ 1}
≥ {| λ ( x) |: x ∈ X , C 2 || x ||≤ 1} = C 2−1 || λ || .
(13)
Hence ψ is one-to-one. To show that ψ is onto, let Λ ∈ ( X × X ) ∗ . Then Λoη ∈ X ′ and
ψ (Λoη ) = Λ .
Theorem 3. 5 Let G be a locally compact group and τ be a topological involution on G. Then
L1 (G,τ ) is amenable if and only if L1 (G ) is amenable.
Proof. Let L1 (G,τ ) be amenable, X be a Banach L1 (G ) -module and ∆ ∈ Z1 (L1 (G), X ∗ ) . If X R
represents X as a real Banach space then it is a real Banach L1 (G,τ ) -module under the module
actions defined by
Iranian Journal of Science & Technology, Trans. A Volume 28, Number A2
Summer 2004
Real group algebras
f .x = ( f + i0).x,
x. f = x.( f + i 0), ( f ∈ L1 (G, τ ), x ∈ X )
295
(14)
Now we define the map D : L1 (G ,τ ) → X R∗ by Df = Re ∆( f + i 0) . Clearly D is a real-linear
mapping and since for each f ∈ L1 (G, τ )
|| Df ||≤ sup{| ∆ ( f + i 0)( x) |: x ∈ X , || x ||≤ 1} ≤|| ∆ || || f || ,
(15)
D is continuous. On the other hand, for each f , g ∈ L1 (G, τ ) ,
( Df ).g = Re(∆( f + i0).( g + i0)), f .( Dg ) = Re(( f + i0).∆( g + i 0)) .
(16)
Hence D ( fg ) = ( Df ).g + f .( Dg ) and so D ∈ Z 1 ( L1 (G , τ ), X R∗ ) . The amenability of L1 (G,τ )
implies that there exists u ∈ X R∗ such that D = δ u . Now we define λ : X → C/
by λ ( x) = u ( x) − iu (ix) . Clearly λ ∈ X ∗ and for f ∈ L1 (G ,τ ), x ∈ X we have
(λ..( f + i 0))( x) = u ( f .x) − iu ( f .ix), (( f + i0).λ )( x) = u ( x. f ) − iu (ix. f ) .
(17)
We can show that ∆ ( f + ig ))( x) = (δ λ ( f + ig ))( x) for every f , g ∈ L1 (G,τ ) and x ∈ X . Hence
∆ = δ λ and so ∆ is an inner derivation, i.e. H 1 ( L1 (G ), X ∗ ) = {0} Thus L1 (G ) is amenable.
Conversely, let L1 (G ) be amenable and let X be a real Banach L1 (G, τ ) -module. By Lemma
3.3 it is enough to show that H 1 ( L1 (G ), X ′) = {0} . Let D : L1 (G, τ ) → X ′ be a continuous real
derivation. By Lemma 3.4, X × X is a Banach space under the norm ||| ( x, y ) ||| = max{|| x ||, || y ||} .
The map ψ : X ′ → ( X × X ) ∗ , defined by ψ (λ )( x, y ) = λ ( x) + iλ ( y )( x, y ∈ X , λ ∈ X ′) is a
continuous real-linear mapping which is one-one and onto. The space X × X is a Banach L1 (G ) module under the familiar module actions. Now we define the map ∆ : L1 (G ) → ( X × X ) ∗ by
∆( f + ig ) = ψ ( Df ) + iψ ( Dg ) . Clearly ∆ is a complex linear mapping and for f , g ∈ L1 (G ) ,
|| ∆ ( f + ig ) ||≤|| ψ || || D || || f ||1 + ψ || D || || g ||≤ 2 || ψ || || D || max{|| f ||1 , || g ||1 }
(18)
≤ 2 || ψ || || D || || f + ig ||1 .
Hence ∆ is continuous. Considering the module actions on X × X we can show that
ψ (( Df ).g ) = ψ ( Df ).( g + i0)
(19)
ψ ( f .( Dg )) = ( f + i0).ψ ( Dg )
(20)
and
Since D is a X ′ -derivation on L1 (G, τ ) , by using the above equation we have
∆ (( f1 + ig 1 ).( f 2 + ig 2 )) = ( ∆ ( f 1 + ig 1 ).( f 2 + ig 2 ) + ( f1 + ig 1 ).(∆ ( f 2 + ig 2 )).
Therefore, ∆ ∈ Z 1 ( L1 (G ), ( X × X ) ∗ ) and so there exists Λ ∈ ( X × X ) ∗ such that ∆ = δ Λ . Since
ψ is onto and one-to-one there exists a unique λ ∈ X ′ such that Λ = ψ (λ ) .
Now we notice that ψ ( f .η ) = ( f + i 0).ψ (η ) and ψ (η. f ) = ψ (η ).( f + i 0) for every
f ∈ L1 (G, τ ) and η ∈ X ′ . Hence
ψ ( Df ) = ψ ( Df ) + iψ ( D0) = ∆( f + i 0) = δ Λ ( f + i0)
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A. Ebadian / A. R. Medghalchi
( f + i0).Λ − Λ.( f + i 0) = ( f + i0).ψ (λ ) − ψ (λ ).( f + i0)
(21)
= ψ ( f .λ ) − ψ (λ. f ) = ψ (δ λ ( f )).
Since ψ is one-to-one, it implies that D( f ) = δ λ ( f ) for each f ∈ L1 (G, τ ) and D = δ λ . This
completes the proof.
Theorem 3. 6. Let G be a locally compact group and let τ be a topological involution on G. Then
L1 (G,τ ) is weakly amenable if and only if L1 (G ) is weakly amenable.
Let L1 (G, τ ) be a weakly amenable real Banach algebra. We show that for each
∆ ∈ Z ( L1 (G ), L1 (G ) ∗ ) there exists Λ ∈ L1 (G ) ∗ such that ∆ = δ Λ . Let η : L1 (G, τ ) → L1 (G ) be
defined by
η ( f ) = f + i 0 and ψ : L1 (G,τ )′ → L1 (G ) ∗ × L1 (G ) ∗ be defined by
ψ (λ )( f + ig ) = λ ( f ) + iλ ( g ). By Lemma 3.4, η and ψ are continuous real-linear mapping. Also,
ψ is a one-to-one and onto mapping from the real Banach space L1 (G,τ )′ onto L1 (G ) ∗ as a real
Banach
space.
By
the
open
Mapping
Theorem
for
real
Banach
spaces,
−1
1
∗
1
∗
1
ψ : L (G ) × L (G ) → L (G , τ )′ is a real-linear continuous mapping. Now if we
define D = ψ −1o∆oη , then it is easy to see that D is a real-linear continuous mapping. To show that
D is an L1 (G, τ )′ -derivation on L1 (G,τ ) we see that
Proof.
1
ψ ( D( fg )) = (ψoD)( fg ) = ∆( fg + i 0) = ∆(( f + i0).( g + i 0))
= (∆oη )( f ).( g + i0) + ( f + i 0).(∆oη )( g )
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= ψ ( Df ).( g + i 0) + ( f + i 0).ψ ( Dg ) .
On the other hand, ψ ( µ )( f + i 0) = ψ ( µ . f ) and ( f + i 0).ψ ( µ ) = ψ (a.µ ) for f ∈ L1 (G ) and
µ ∈ L1 (G,τ )′ . Hence, for each f , g ∈ L1 (G,τ ) we have
ψ ( D( fg )) = ψ ( Df .g ) + ψ ( f .Dg ) = ψ ( Df .g + f .Dg ).
ψ
(23)
D is an L1 (G,τ )′ -derivation, i.e.
we conclude that
D ∈ Z ( L (G, τ ), L (G, τ )′) . By Lemma 3.3, the weak amenability of L1 (G,τ ) implies that there
exists λ ∈ L1 (G, τ ) ′ such that D = δ λ . By definition of D and the above equalities it implies that
∆ = δ ψ ( λ ) , and so L1 (G ) is weakly amenable.
Conversely, let L1 (G ) be weakly amenable and D ∈ Z 1 ( L1 (G, τ ), L1 (G, τ ) ′) . By Lemma 3.4
the mapψ : L1 (G, τ )′ → L1 (G ) ∗ × L1 (G ) ∗ , defined by ψ (λ )( f + ig ) = λ ( f ) + iλ ( g ) , is a reallinear continuous one-to-one mapping onto L1 (G ) ∗ × L1 (G ) ∗ , as a real Banach space.
Now we define the map ∆ : L1 (G) → L1 (G)∗ × L1 (G)∗ by ∆ ( f + ig ) = ψ ( Df ) + iψ ( Dg ). Similar
to the proof of Theorem 3.4 we can show that ∆ is a continuous derivation. Hence there exists
Λ ∈ L1 (G ) ∗ × L1 (G ) ∗ such that ∆ = δ Λ . Since ψ is one-to-one and onto, there exists a unique
λ ∈ L1 (G,τ )′ such that Λ = ψ (λ ) . It can be shown that D = δ λ and so L1 (G,τ ) is weakly
amenable by Lemma 3.3.
Since
1
1
is
one-one,
1
Corollary 3. 7. Let G be a locally compact group and τ be a topological involution on G . Then
(i) L1 (G,τ ) is amenable if and only if G is amenable.
(ii) L1 (G,τ ) is weakly amenable.
Iranian Journal of Science & Technology, Trans. A Volume 28, Number A2
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297
Real group algebras
Proof. By Theorem 3.5 and 3.6 the amenability and weak amenability of L1 (G,τ ) and L1 (G ) are
equivalent. Since L1 (G ) is amenable if and only if G is amenable [5], (i) follows. Since L1 (G ) is
weakly amenable [6], we conclude that L1 (G,τ ) is also weakly amenable.
4. MULTIPLIERS
In this section we characterize the multipliers of L1 (G,τ ) . A bounded real linear operator T on
L1 (G,τ ) is called a left (right) multiplier if T ( f ∗ g ) = (Tf ) ∗ g (= f ∗ Tg ) f , g ∈ L1 (G, τ )) .
Definition 4. 1. Let δ x be the point mass at x ∈ G . We define m x =
δ x + δ τ ( x)
2
R x ( f ) = m x ∗ f ( f ∈ L1 (G , τ )).
It is clear that m x oτ = m x (since τ 2 = τ ) and || m x ||= 1 . Therefore, m x ∈ M (G ,τ ) .
and
Lemma 4. 2. Let µ be a measure in M (G ) such that f ∗ µ ∈ L1 (G,τ ) for every f ∈ L1 (G,τ ) .
Then µ ∈ M (G, τ ) .
Proof. We have f ∗ µ ( x ) =
∫
G
f ( xy −1 ) dµ (( y ) ( x ∈ G ) . Therefore,
f ∗ ( µ oτ )( x) = ∫ f ( xy −1 ) d ( µ oτ )( y )
G
= ∫ f ( x (τ ( y −1 )) dµ ( y )
G
=
∫
G
=
∫
G
f (τ ( x ) y −1 ) d µ ( y )
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f (τ ( x ) y −1 ) dµ ( y )
= f ∗ µ (τ ( x)) = f ∗ µ ( x) .
So f ∗ ( µ oτ ) = f ∗ µ for every f in L1 (G,τ ) . Since L1 (G,τ ) has a bounded approximate
identity, we have µ oτ = µ . Hence µ ∈ M (G, τ ) .
Theorem 4. 3. Let T be a left multiplier on L1 (G,τ ) . Then there exists a unique µ ∈ M (G,τ )
such that Tf = f ∗ µ ( f ∈ L1 (G, τ )) and || µ ||=|| T || .
Proof. We define T0 : L1 (G ) → L1 (G ) by T0 ( f ) = T ( g ) + iT (h) where f = g + ih . We have
T0 ( f1 ∗ f 2 ) = T0 (( g1 + ih1 ) ∗ ( g 2 + ih2 )
= T0 ( g1 ∗ g 2 + ih1 ∗ g 2 + ig1 ∗ h2 − h1 ∗ h2 )
= T0 ( g1 ∗ g 2 − h1 ∗ h2 ) + iT (h1 ∗ g 2 + g1 ∗ h2 )
(25)
= g1 ∗ Tg 2 − h1 ∗ Th2 + ih1 ∗ Tg 2 + ig 1 ∗ Th2
= ( g1 + ih1 ) ∗ (Tg 2 + iTh2 )
= f1 ∗ T0 f 2 .
Summer 2004
Iranian Journal of Science & Technology, Trans. A, Volume 28, Number A2
298
A. Ebadian / A. R. Medghalchi
Hence, T0 is a left multiplier on L1 (G ) . Therefore, by [7] there exists a unique µ ∈ M (G, τ ) such
that T0 h = h ∗ µ for every h in L1 (G ) and || T0 ||=| µ | (G ) . Consequently, Tf = f ∗ µ for every
L1 (G,τ ) and || T ||≤|| T0 ||=| µ | (G ) . Now, since f ∗ µ ∈ L1 (G,τ ) for all f ∈ L1 (G,τ ) ,
µ ∈ M (G,τ ) by Lemma 4.2. But by the proof of [7, Theorem 1] we have
µ = ω ∗ − limα T0 eα = ω ∗ − limα Teα .Now, since || Teα ||≤|| T || ,
| µ | (G ) ≤|| T || .
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Therefore, || T ||=|| µ || .
REFERENCES
1.
2.
3.
4.
5.
6.
7.
Ingelstam, L. (1964). Real Banach algebras, Ark. Math. 5, 239-270.
Kulkarni, S. H. & Limaye, B. V. (1992). “Real function algebras” Marcel Dekker, Inc.
Hewitt, E. & Ross, K. A. (1963) & (1970). Abstract harmonic analysis, Vols, I, II Springer-Verlag, Berlin.
Bonsall, F. F. & Duncan, J. (1973). “Complete normed algebras”, Springer-Verlag, New.
Johnson, B. E. (1972). Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127.
Johnson, B. E. (1991). Weak amenability of group algebras, Bull. London. Math. Soc. 23, 281-284.
Wendel, J. G. (1952). Left centralizers and isomorphism of group algebras, Pacific. J.Math. 2, 251-261.
Iranian Journal of Science & Technology, Trans. A Volume 28, Number A2
Summer 2004