A common fixed point property for semigroups is applied to show that the
group algebra $L^1(G)$ of a locally compact group $G$ is $2m$-weakly amenable
for each integer $m\geq 1$.Comment: J. Math Anal. Appl. to appea

Zhang, Yong

English

arXiv

AbstractA common fixed point property for semigroups is applied to show that the group algebra L1(G) of a locally compact group G is 2m-weakly amenable for each integer m≥1

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2m-Weak amenability of group algebras 

J. Math. Anal. Appl. 396 (2012) 412–416
Contents lists available at SciVerse ScienceDirect
Journal of Mathematical Analysis and
Applications
journal homepage: www.elsevier.com/locate/jmaa
2m-Weak amenability of group algebras
Yong Zhang
Department of Mathematics, University of Manitoba, Winnipeg MB R3T 2N2, Canada
a r t i c l e i n f o
Article history:
Received 1 December 2011
Available online 28 June 2012
Submitted by Tomas Dominguez-Benavides
Keywords:
Derivation
2m-weak amenability
Locally compact group
Common fixed point
L-embedded
Semigroup
Weakly almost periodic
a b s t r a c t
A common fixed point property for semigroups is applied to show that the group algebra
L1(G) of a locally compact group G is 2m-weakly amenable for each integerm ≥ 1.
© 2012 Elsevier Inc. All rights reserved.
1. Introduction
Let A be a Banach algebra and X a Banach A-bimodule. A linear mapping D: A → X is called a derivation if it satisfies
D(ab) = aD(b)+D(a)b (a, b ∈ A). Given any x ∈ X , the mapping adx: a → ax−α(a ∈ A) is a continuous derivation, called
an inner derivation.
If X is a BanachA-bimodule, then the dual space X∗ of X is naturally a BanachA-bimodule with theA-module actions
defined by
⟨x, af ⟩ = ⟨xa, f ⟩ ⟨x, fa⟩ = ⟨ax, f ⟩ (a ∈ A, f ∈ X∗, x ∈ X).
Note that the Banach algebraA itself is a BanachA-bimodule with the product giving the module actions. SoA(n), the n-th
dual space ofA, is naturally a BanachA-bimodule in the above sense for each n ∈ N. The Banach algebraA is called n-weakly
amenable if every continuous derivation fromA intoA(n) is inner. IfA is n-weakly amenable for each n ∈ N then it is called
permanently weakly amenable.
Let G be a locally compact group. The integral of a function f on a measurable subset K of G against a fixed left Haar
measure is denoted by

K fdx. Two functions on G are regarded identical if they are equal to each other almost everywhere
with respect to the left Haar measure. The group algebra L1(G) is the Banach algebra consisting of all absolutely integrable
functions on G (with respect to the left Haar measure), equipped with the convolution product and the usual L1 norm
∥f ∥1 :=

G
|f (t)|dt.
When G is discrete, L1(G) is ℓ1(G) consisting of all absolutely summable functions on G.
Johnson showed in [1] that L1(G) is always 1-weakly amenable for any locally compact group G. It was shown further
in [2] that L1(G) is in fact n-weakly amenable for all odd numbers n. Whether this is still true for even numbers n was left
E-mail address: zhangy@cc.umanitoba.ca.
0022-247X/$ – see front matter© 2012 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmaa.2012.06.037
Y. Zhang / J. Math. Anal. Appl. 396 (2012) 412–416 413
open in [2]. For a free group G, Johnson proved later in [3] that ℓ1(G) is indeed 2m-weakly amenable for any m ∈ N. The
problem has been resolved affirmatively for general locally compact group G in [4] and in [5] independently, using a theory
established in [6].
In this note we present a short proof to the n-weak amenability of L1(G) for even numbers n. Our proof is based on
a common fixed point property for semigroups. In Section 2 we study this fixed point property. For the general theory
concerning amenability and fixed point properties of locally compact groups we refer the reader to [7,8]. The proof to the
main result will be given in Section 3.
2. Common fixed points for semigroups
Let S be a (discrete) semigroup. The space of all bounded complex valued functions on S is denoted by ℓ∞(S). It is a
Banach space with the uniform supremum norm. In fact ℓ∞(S) = (ℓ1(S))∗, the dual space of ℓ1(S). For each s ∈ S and each
f ∈ ℓ∞(S) let ℓsf be the left translate of f by s, that is ℓsf (t) = f (st) (t ∈ S) (the right translate rsf is defined similarly).
A function f ∈ ℓ∞(S) is called weakly almost periodic if its left orbit LO(f ) = {ℓsf : s ∈ S} is relatively compact in the
weak topology of ℓ∞(S). The space of all weakly almost periodic functions on S is denoted by WAP(S), which is a closed
subspace of ℓ∞(S) containing the constant function and invariant under the left and right translations. A linear functional
m ∈ WAP(S)∗ is a mean onWAP(S) if ∥m∥ = m(1) = 1. A mean m onWAP(S) is a left invariant mean (abbreviated as LIM)
ifm(ℓsf ) = m(f ) for all s ∈ S and all f ∈ WAP(S). If S is a group, it is well known thatWAP(S) always has a LIM [7].
Let X be a Banach space and C a nonempty subset of X . A mapping T : C → C is called nonexpansive if ∥T (x) − T (y)∥ ≤
∥x− y∥ for all x, y ∈ C . When X is a separable locally convex topological space whose topology is determined by a family Q
of seminorms on X , we will denote it by (X,Q ) to highlight the topology Q .
Let C be a subset of a locally convex topological vector space (X,Q ). We say that S = {Ts : s ∈ S} is a representation
of S on C if for each s ∈ S, Ts is a mapping from C into C and Tst(x) = Ts(Ttx) (s, t ∈ S, x ∈ C). The representation is
called continuous if each Ts (s ∈ S) is Q–Q continuous; It is called equicontinuous if for each neighborhoodN of 0 there is
a neighborhood O of 0 such that Ts(x) − Ts(y) ∈ N whenever x, y ∈ C, x − y ∈ O and s ∈ S. The representation is called
affine if C is convex and each Ts (s ∈ S) is an affine mapping, that is Ts(ax+ by) = aTs(x)+ bTs(y) for all constants a, b ≥ 0
with a+ b = 1, s ∈ S and x, y ∈ C . We say that x ∈ C is a common fixed point for (the representation of) S if Ts(x) = x for all
s ∈ S.
The following fixed point theorem was proved in [9].
Theorem 1. Let S be a discrete semigroup andS an equicontinuous affine representation of S on aweakly compact convex subset
C of a separated locally convex space X. If WAP(S) has a left invariant mean then C contains a common fixed point for S.
Let B be a nonempty bounded subset of a Banach space X . By definition the Chebyshev radius of B in X is
rB = inf

r ≥ 0 : ∃x ∈ X sup
b∈B
∥x− b∥ ≤ r

.
Clearly we have 0 ≤ rB <∞ and
sup
b∈B
∥x− b∥ ≥ rB for each x ∈ X . (2.1)
The Chebyshev center of B in X is defined to be
CB =

x ∈ X : sup
b∈B
∥x− b∥ ≤ rB

.
Chebyshev center has been extensively used in the field of fixed point theory (see [10,11]). Some asymptotic version of it
has been employed to study fixed point properties of semigroups [12–14].
We now recall that a Banach space X is L-embedded if the image of X under the canonical embedding into its bidual
X∗∗, still denoted by X , is an ℓ1 summand in X∗∗, that is if there is a subspace Xs of X∗∗ such that X∗∗ = X ⊕1 Xs, where
⊕1 denotes the ℓ1 direct sum. The class of L-embedded Banach spaces includes all L1(Σ, µ) (the space of all absolutely
integrable functions on a measure space (Σ, µ)), preduals of von Neumann algebras, dual spaces of M-embedded Banach
spaces and the Hardy space H1. In particular, given a locally compact group G, the space L1(G) is L-embedded. So are its even
duals L1(G)(2m) (m ∈ N). We refer to [15] for more details of the theory concerning this type of Banach spaces. We also refer
to [16–18] for the study of fixed points of various mappings in an L-embedded Banach space. In [16], as an application of
a fixed point theorem, a surprising short solution to the well-known derivation problem was given. The problem was first
settled by Losert in [6].
We now give a common fixed point theorem for semigroups, which will provide the major machinery for our proof to
the main result.
Theorem 2. Let S be a discrete semigroup andS a representation of S on an L-embedded Banach space X as nonexpansive affine
mappings. Suppose that WAP(S) has a LIM and suppose that there is a nonempty bounded set B ⊂ X such that B ⊆ Ts(B) for all
s ∈ S, then X contains a common fixed point for S.
414 Y. Zhang / J. Math. Anal. Appl. 396 (2012) 412–416
Proof. We use the idea of [16] to show that there is a nonempty weakly compact convex set in X that is S-invariant. We
first regard B as a subset of X∗∗. Let rB be the Chebyshev radius and C the Chebyshev center of B in X∗∗. Then C is nonempty,
weak* compact and convex. In fact, for each r > rB,
Cr :=

x ∈ X∗∗ : sup
b∈B
∥x− b∥ ≤ r

is nonempty by the definition of rB. Note that Cr = ∩b∈B B[b, r], where B[b, r] denotes the closed ball in X∗∗ centered at b
with radius r . The set Cr is convex and weak* compact since each B[b, r] is. The collection {Cr : r > rB} is decreasing as
r decreases. Thus C = ∩r>rB Cr is nonempty and is still weak* compact and convex. By the L-embeddedness of X there is
a subspace Xs of X∗∗ such that X∗∗ = X ⊕1 Xs. Let x ∈ C . then there are c ∈ X and ξ ∈ Xs such that x = c + ξ . For each
b ∈ B, ∥x− b∥ = ∥c − b∥ + ∥ξ∥. So
rB ≥ sup
b∈B
∥x− b∥ = sup
b∈B
∥c − b∥ + ∥ξ∥ ≥ rB + ∥ξ∥.
The last inequality is due to (2.1). Therefore, we must have ξ = 0. This shows that C ⊂ X . The weak* compactness of C (in
X∗∗) is the same as the weak compactness of it (in X). So C is a nonempty, weakly compact and convex subset of X .
Now for s ∈ S, b ∈ B and x ∈ C we have
∥Ts(x)− Ts(b)∥ ≤ ∥x− b∥ ≤ rB
since Ts is nonexpansive. This implies that ∥Ts(x) − a∥ ≤ rB for a ∈ Ts(B) (s ∈ S, x ∈ C). In particular, this holds for
all a ∈ B since B ⊆ Ts(B). Thus Ts(x) ∈ C whenever x ∈ C and s ∈ S, showing that C is S-invariant. Note that a
nonexpansive representation of S is indeed equicontinuous. By Theorem 1, there is a common fixed point for S in C . The
proof is complete. 
Theorem 1 has been extended to the general semitopological semigroup setting in [19]. A more general version of
Theorem 2 and some discussion on when there is a set B such that Ts(B) = B for all s ∈ S can also be found there.
3. 2m-weak amenability of L1(G)
Let X be a Banach space. Denote the space of all bounded linear operators on X by B(X). The space B(X) is a Banach
algebra with the operator norm topology and the composition product. So is B(X)× B(X)op with the product topology and
coordinatewise operations, where B(X)op is the algebra formed by reversing the order of the product in B(X). The strong
operator topology (or briefly so-topology) on B(X)× B(X)op is the topology induced by the family of seminorms {px : x ∈ X},
where
px(S, T ) = max{∥S(x)∥, ∥T (x)∥} (S, T ∈ B(X))
(see [20, p. 327]).
Given a locally compact group G, let M(G) be the space of all bounded complex valued regular Borel measures on G.
With the convolution product of measures and with the norm induced by the total variation, M(G) is a Banach algebra
containing L1(G) as a closed ideal. In fact, M(G) is the multiplier algebra of L1(G), and as the multiplier algebra of L1(G),
M(G) is a subalgebra of B(L1(G))× B(L1(G))op with each µ ∈ M(G) being identified with (the double multiplier) (ℓµ, rµ) ∈
B(L1(G)) × B(L1(G))op, where ℓµ and rµ denote, respectively, the left multiplier operator and the right multiplier operator
on L1(G) implemented by µ. We refer to [20] for the standard theory about multipliers and multiplier algebras.
It is well-known that lin{δt : t ∈ G}, the linear space generated by the point measures δt (t ∈ G), is dense in M(G) in
the so-topology [20, Proposition 3.3.41(i)]. In particular, for each h ∈ L1(G) there is a net (uα) ⊂ lin{δt : t ∈ G} such that
∥(uα − h) ∗ a∥1 → 0 and ∥a ∗ (uα − h)∥1 → 0 for all a ∈ L1(G).
Recall that ifA is a Banach algebra, then its bidualA∗∗ is a Banach algebra equipped with the Arens product  defined
⟨f , uv⟩ = ⟨v · f , u⟩, v · f ∈ A∗ : ⟨a, v · f ⟩ = ⟨fa, v⟩
for u, v ∈ A∗∗, f ∈ A∗ and a ∈ A. If X is a BanachA-bimodule, then its bidual X∗∗ is naturally a BanachA∗∗-bimodule with
the module actions given by
⟨F , u ·M⟩ = ⟨M · F , u⟩, M · F ∈ A∗ : ⟨a,M · F⟩ = ⟨F · a,M⟩
and
⟨F ,M · u⟩ = ⟨u · F ,M⟩, u · F ∈ X∗ : ⟨x, u · F⟩ = ⟨F · x, u⟩,
F · x ∈ A∗ : ⟨a, F · x⟩ = ⟨x · a, F⟩
for u ∈ A∗∗,M ∈ X∗∗, F ∈ X∗, x ∈ X and a ∈ A. In particular, for any integerm ∈ N,A(2m) is a BanachA∗∗-bimodule.
Y. Zhang / J. Math. Anal. Appl. 396 (2012) 412–416 415
A BanachA-bimodule X is called neo-unital if X = AXA, that is every element x ∈ X may be written in the form x = ayb
for some a, b ∈ A and y ∈ X . IfA has a bounded approximate identity (eα) and X is a neo-unital BanachA-bimodule, then
we may extend theA bimodule actions on X toM(A), the multiplier algebra ofA. The extension is defined as follows.
µx = lim
α
(µeα)x = (µa)yb, xµ = lim
α
x(eαµ) = ay(bµ)
for µ ∈ M(A) and x = ayb ∈ X . Here we note that µa, bµ ∈ A since A is (always) an ideal of M(A). These operations
make X a unital BanachM(A)-bimodule. In this case a continuous derivation D:A→ X∗ may be extended to a continuous
derivation fromM(A) to X∗ by defining
D(µ) = wk*- lim
α
D(µeα) (µ ∈ M(A)).
Moreover this extended D is so-weak* continuous. In fact, if µα → µ in M(A) in the so-topology and x = ayb ∈ X for
a, b ∈ A and y ∈ X , then
lim
α
⟨x,D(µα)⟩ = lim
α
⟨ay,D(bµα)⟩ − lim
α
⟨µαay,D(b)⟩
= ⟨ay,D(bµ)⟩ − ⟨µay,D(b)⟩ = ⟨x,D(µ)⟩.
We refer to the seminar paper [21] and the monograph [20] for more details of the above extensions.
We now can prove the main result of the paper.
Theorem 3. Let G be a locally compact group. Then the group algebra L1(G) is 2m-weakly amenable for each m ∈ N.
Proof. DenoteA = L1(G), X = A(2m) and Y = A(2m−1). Then, as we have indicated, X is a BanachA∗∗-bimodule. Let (eα)
be a bounded approximate identity ofA and let E be a weak* cluster point of (eα) inA∗∗. Then Ea = aE = a for all a ∈ A.
We have theA-bimodule decomposition X = X1 ⊕ X2 ⊕ X3, where
X1 = ℓE ◦ rE(X), X2 = (I − rE)(X), X3 = (I − ℓE) ◦ rE(X).
Here I denotes the identity operator, ℓE is the left multiplication by E and rE the right multiplication by E. It is readily seen
that
X2 = (AY )⊥ ∼= (Y/AY )∗, X1 ⊕ X3 = rE(X) ∼= (AY )∗
as BanachA-bimodules. Similarly, in (AY )∗
(I − ℓE)

(AY )∗
 = (AYA)⊥ ∼= (AY/AYA)∗
and
ℓE

(AY )∗
 ∼= (AYA)∗
as BanachA-bimodules. We have
X3 ∼= (AY/AYA)∗ and X1 ∼= (AYA)∗.
Let D:A→ X be a continuous derivation. Then D = D1 + D2 + D3, where
D1 = ℓE ◦ rE ◦ D : A→ X1, D2 = (I − rE) ◦ D : A→ X2,
D3 = (I − ℓE) ◦ rE ◦ D : A→ X3.
Since ℓE and rE are A-bimodule morphisms, D1,D2 and D3 are continuous derivations. Note that the left A-module action
on Y/AY and the right A-module action on A/AYA are trivial. From [21, Proposition 1.5], D2 and D3 are inner. We now
show that D1 is also inner. Then Dmust be inner.
SinceAYA is neo-unital, we may extend D1 to a continuous derivation fromM(G), the multiplier ofA, to X1. So we may
consider∆: G → X1 ⊂ X defined by
∆(t) = D1(δt) · δt−1 (t ∈ G).
It is readily seen that
∆(ts) = δt ·∆(s) · δt−1 +∆(t) (t, s ∈ G). (3.1)
Let B = ∆(G). Then B is a nonempty bounded subset of X . For each t ∈ G, let Tt be the self mapping on X defined by
Tt(x) = δt · x · δt−1 +∆(t) (x ∈ X).
Using (3.1) onemay check thatS = {Tt : t ∈ G} defines a representation of G on X which is clearly nonexpansive and affine.
Moreover, Tt(∆(s)) = ∆(ts) (t, s ∈ G) and Te = I . Since G is a group, the above implies Tt(B) = B for each t ∈ G. Here G is
regarded as a discrete group.
416 Y. Zhang / J. Math. Anal. Appl. 396 (2012) 412–416
SinceWAP(G) has a LIM and X is L-embedded, by Theorem 2, there is ξ ∈ X such that
δt · ξ · δt−1 +∆(t) = ξ for all t ∈ G.
So D1(δt) = ξ · δt − δt · ξ = ad−ξ (δt) (t ∈ G). Let x = ℓE ◦ rE(−ξ). Then x ∈ X1. Also D1(δt) ∈ X1. For any ayb ∈ AYAwith
a, b ∈ A and y ∈ Y , we have
⟨ayb,D1(δt)⟩ = ⟨ayb · δt − δt · ayb,−ξ⟩ = ⟨E(ayb · δt − δt · ayb)E,−ξ⟩
= ⟨ayb · δt − δt · ayb, x⟩ = ⟨ayb, adx(δt)⟩ t ∈ G.
So it is true that D1(δt) = adx(δt) for all t ∈ G. From what we have shown before stating the current theorem, both D1
and adx, as continuous derivations from M(G) into the dual of a neo-unital A-bimodule, are so-weak* continuous. Since
lin(δt : t ∈ G) is dense inM(G) in the so-topology, we finally have
D1(f ) = adx(f ) (f ∈ A = L1(G)),
therefore D1 is inner. The proof is complete. 
Acknowledgment
This work was supported by NSERC 238949-2011.
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