# Multinorms and Approximate Amenability of Weighted Group Algebras

Authors: Saman Ghaderkhani
Journal: Chinese Journal of Mathematics
 (2014)

---

## Abstract

LetG be a locally compact group, and take p
,
q with 1
≤
p
,
q
<
∞. We prove that, for any left (
p
,
q
)-multiinvariant functional on L
∞
(
G
) and for any weight function ω
≥
1 on G, the approximate amenability of the Banach algebra L
1
(
G
,
ω
) implies the left (
p
,
q
)-amenability of G, but in general the opposite is not true. Our proof uses the notion of multinorms. We also investigate the approximate amenability of M
(
G
,
ω
).


---

## Body

## 1. Introduction

Ghahramani and Loy in [1] introduced the notion of approximately amenable Banach algebras and, among other interesting results, they proved that for a locally compact group G, the group algebra L
1
(
G
) is approximately amenable if and only if G is amenable.LetG be a locally compact group. Recently Dales et al. in [2] defined another generalized notion of amenability, called left (
p
,
q
)-amenability of G, for any p, q such that 1
≤
p
≤
q
<
∞. The aim of the present paper is to show that, for any left (
p
,
q
)-multiinvariant function on L
∞
(
G
) and for any weight function ω
≥
1 on G, the approximate amenability of the Banach algebra L
1
(
G
,
ω
) implies the left (
p
,
q
)-amenability of G, but in general the opposite is not true.


## 2. Notation and Preliminaries

In this section, we will recall various notations that we will use and give the definitions and some conventions.


### 2.1. Weighted Group Algebras and Approximate Amenability of Banach Algebras

LetG denote a locally compact group with a fixed left Haar measurable λ and ω a weight function on G; that is, a Borel measurable function ω
:
G
→
ℝ
+ such that
(1)
ω
(
x
·
y
)
≤
ω
(
x
)
ω
(
y
)
,
(
x
,
y
∈
G
)
.
The weighted group algebra L
1
(
G
,
ω
) is the space of all measurable complex-valued functions f on G such that
(2)
∫
G
‍
|
f
(
x
)
|
ω
(
x
)
d
x
<
∞
and is equipped with the convolution product * of functions; that is, for f
,
g
∈
L
1
(
G
,
ω
) and x
∈
G,
(3)
(
f
*
g
)
(
x
)
=
∫
G
‍
f
(
x
y
-
1
)
g
(
y
)
d
y
,
and the norm
(4)
∥
f
∥
ω
=
∫
G
‍
|
f
(
x
)
|
ω
(
x
)
d
x
.
Also, let L
∞
(
G
,
ω
-
1
) be the space of all measurable complex-valued functions ϕ on G, such that ϕ
/
ω is essentially bounded, and for ϕ
∈
L
∞
(
G
,
ω
-
1
) define
(5)
∥
ϕ
∥
∞
,
ω
=
∥
ϕ
ω
∥
∞
=
ess sup
{
|
ϕ
(
x
)
ω
(
x
)
|
∣
x
∈
G
}
.
The spaces L
1
(
G
,
ω
) and L
∞
(
G
,
ω
-
1
) are in duality by
(6)
〈
f
,
ϕ
〉
=
∫
G
‍
f
ϕ
d
λ
(
f
∈
L
1
(
G
,
ω
)
,
ϕ
∈
L
∞
(
G
,
ω
-
1
)
)
,
and if ω
(
x
)
≥
1, (
x
∈
G
), then L
∞
(
G
)
⊆
L
∞
(
G
,
ω
-
1
). Let M
(
G
,
ω
) be the Banach space of all complex-valued, regular Borel measures μ on G such that
(7)
∥
μ
∥
ω
=
∫
G
‍
ω
(
x
)
d
|
μ
|
(
x
)
<
∞
.
Note that M
(
G
,
1
)
=
M
(
G
). If ω
(
x
)
≥
1, (
x
∈
G
), then M
(
G
,
ω
) is a subalgebra of M
(
G
).Let𝒜 be a Banach algebra and X a Banach 𝒜-bimodule. A bounded linear map D
:
𝒜
→
X is called a derivation if
(8)
D
(
a
b
)
=
D
(
a
)
·
b
+
a
·
D
·
(
b
)
(
a
,
b
∈
𝒜
)
.
For every x
∈
X we define a
d
x
𝒜 by
(9)
a
d
x
𝒜
(
a
)
=
a
x
-
x
a
(
a
∈
𝒜
)
.
It is easily seen that a
d
x
𝒜 is a derivation. Derivation of this form is called inner derivation.Let𝒜 be a Banach algebra and X a Banach 𝒜-bimodule. Then X
*, the dual space of X, is a Banach 𝒜-bimodule for operations given by
(10)
〈
x
,
a
ξ
〉
=
〈
x
a
,
ξ
〉
,
〈
x
,
ξ
a
〉
=
〈
a
x
,
ξ
〉
111111111
(
a
∈
𝒜
,
x
∈
X
,
ξ
∈
X
*
)
.
X
* is the dual module of X; and in particular 𝒜
* is the dual module of 𝒜.A Banach algebra𝒜 is called amenable if, for any 𝒜-bimodule X, any derivation D
:
𝒜
→
X
* is inner. This definition of amenability was introduced by Johnson in [3]. A Banach algebra 𝒜 is called weakly amenable if any derivation D
:
𝒜
→
𝒜
* is inner. Trivially, any amenable Banach algebra is weakly amenable.LetG be a locally compact group; a mean on L
∞
(
G
) is a positive functional m
∈
L
∞
(
G
)
* such that
(11)
〈
1
,
m
〉
=
∥
m
∥
=
1
.
A mean m on L
∞
(
G
) is called left invariant if
(12)
〈
δ
x
*
ϕ
,
m
〉
=
〈
ϕ
,
m
〉
(
x
∈
G
,
ϕ
∈
L
∞
(
G
)
)
.
A locally compact group G is called amenable if there exists a left invariant mean on L
∞
(
G
). Note that G is amenable if and only if L
1
(
G
) is an amenable Banach algebra.A derivationD
:
𝒜
→
X is called approximately inner if there exists net (
ξ
a
)
⊆
X such that, for every a
∈
𝒜,
(13)
D
(
a
)
=
norm-
lim
⁡
a
⁡
(
a
·
ξ
a
-
ξ
a
·
a
)
.
Recall from [1] that a Banach algebra 𝒜 is called approximately amenable if, for any 𝒜-bimodule X, any derivation D
:
𝒜
→
X
* is approximately inner.


### 2.2. Banach Spaces

Forn
∈
ℕ
=
{
1,2
,
…
}, we set ℕ
n
=
{
1
,
…
,
n
}. Let E be a normed space. For each k
∈
ℕ, we denote by E
k the linear space direct product of k copies of E. The closed unit ball of E is denoted by B
1
(
E
). We denote the dual of E by E
*; the action of λ
∈
E
* on an element x
∈
E is written as 〈
x
,
λ
〉.Following the notation of [4, 5] we define the weak
p
-summing norm (for 1
≤
p
<
∞) on E
n by
(14)
μ
p
,
n
(
x
)
=
sup
⁡
{
(
∑
i
=
1
n
‍
|
〈
x
i
,
λ
〉
|
p
)
1
/
p
:
λ
∈
B
1
(
E
*
)
}
,
where x
=
(
x
1
,
…
,
x
n
)
∈
E
n. See also [6, 7]. Notice that, by the weak* density of B
1
(
E
) in B
1
(
E
*
*
), the weak p-summing norm on (
E
*
)
n can also be computed as
(15)
μ
p
,
n
(
λ
)
=
sup
⁡
{
(
∑
i
=
1
n
‍
|
〈
x
,
λ
i
〉
|
p
)
1
/
p
:
x
∈
B
1
(
E
)
}
,
where λ
=
(
λ
1
,
…
,
λ
n
)
∈
(
E
*
)
n.


### 2.3. Multinormed Spaces

The following definition is by Dales and Polyakov. For a full account of the theory of multinormed spaces, see [4].Definition 1.
Let(
E
,
∥
·
∥
) be a normed space, and let (
∥
·
∥
n
:
n
∈
ℕ
) be a sequence such that ∥
·
∥
n is a norm on E
n for each n
∈
ℕ, with ∥
·
∥
1
=
∥
·
∥ on E. Then the sequence (
∥
·
∥
n
:
n
∈
ℕ
) is a multinorm if the following axioms hold (where in each case the axiom is required to hold for all n
≥
2 and all x
1
,
…
,
x
n
∈
E): (A1)
∥
x
σ
(
1
)
,
…
,
x
σ
(
n
)
∥
n
=
∥
(
x
1
,
…
,
x
n
)
∥
n for each permutation σ of ℕ
n;
(A2)
∥
(
α
1
x
1
,
…
,
α
n
x
n
)
∥
n
≤
max
⁡
i
∈
ℕ
n
|
α
i
|
∥
(
x
1
,
…
,
x
n
)
∥
n  (
α
1
,
…
,
α
n
)
∈
ℂ;
(A3)
∥
(
x
1
,
…
,
x
n
-
1
,
0
)
∥
n
=
∥
(
x
1
,
…
,
x
n
-
1
)
∥
n
-
1;
(A4)
∥
(
x
1
,
…
,
x
n
-
2
,
x
n
-
1
)
∥
n
=
∥
(
x
1
,
…
,
x
n
-
2
,
x
n
-
1
)
∥
n
-
1.
The normed spaceE equipped with a multinorm is a multinormed space, denoted in full by (
(
E
n
,
∥
·
∥
n
)
:
n
∈
ℕ
). We say that such a multinorm is based on E.Definition 2.
Let(
(
E
n
,
∥
·
∥
n
:
n
∈
ℕ
) be a multinormed space. A subset B
⊂
E is multibounded if
(16)
mb
(
B
)
:
=
sup
⁡
{
∥
(
x
1
,
…
,
x
n
)
∥
n
:
x
1
,
…
,
x
n
∈
B
,
n
∈
ℕ
}
<
∞
.
The constant mb
(
B
) is the multibound of B.The following easy remark is from [4, Proposition 6.5
(
ii
)].Remark 3.
LetE be a multinormed space. Then the absolutely convex hull of a multibounded set is multibounded, with the same multibound.


### 2.4. The(
p
,
q
)-Multinorm

Following [4], we now introduce an important class of multinorms. Let E be a normed space, and take p
,
q with 1
≤
p, q
<
∞. For each n
∈
ℕ and each x
=
(
x
1
,
…
,
x
n
)
∈
E
n, we define
(17)
∥
x
∥
n
(
p
,
q
)
=
sup
⁡
{
(
∑
i
=
1
n
|
〈
x
i
,
λ
i
〉
|
q
‍
)
1
/
q
:
11111111
λ
=
(
λ
1
,
…
,
λ
n
)
∈
(
E
*
)
n
,
μ
p
,
n
(
λ
)
≤
1
(
∑
i
=
1
n
|
〈
x
i
,
λ
i
〉
|
q
‍
)
1
/
q
}
.
It is clear that ∥
·
∥
n
(
p
,
q
) is a norm on E
n. As proved in [4], in the case where 1
≤
p
≤
q
<
∞, the sequence (
∥
·
∥
n
(
p
,
q
)
:
n
∈
ℕ
) is a multinorm based on E.Definition 4.
LetE be a normed space, and take p
,
q with 1
≤
p
≤
q
<
∞. Then the multinorm (
∥
·
∥
n
(
p
,
q
)
:
n
∈
ℕ
) described above is the (
p
,
q
)-multinorm over E.
A subset ofE is (
p
,
q
)-multibounded if it is multibounded with respect to the (
p
,
q
)-multinorm. The (
p
,
q
)-multibound of such a set B is denoted by m
b
p
,
q
(
B
).Lemma 5.
LetE be a normed space, and take p
,
q with 1
≤
p, q
<
∞. Then, for each n
∈
ℕ and λ
=
(
λ
1
,
…
,
λ
n
)
∈
(
E
*
)
n, we have
(18)
∥
λ
∥
n
(
p
,
q
)
=
sup
⁡
{
(
∑
i
=
1
n
‍
|
〈
x
i
,
λ
i
〉
|
q
)
1
/
q
:
hhhhhhhh
x
=
(
x
1
,
…
,
x
n
)
∈
(
E
*
)
n
,
μ
p
,
n
(
x
)
≤
1
(
∑
i
=
1
n
‍
|
〈
x
i
,
λ
i
〉
|
q
)
1
/
q
}
.Proof.
This is proved in [4, Proposition 4.10]; it follows from the Principal of Local Reflexivity.


### 2.5. Left(
p
,
q
)-Multiinvariant Means

Definition 6.
LetG be a locally compact group, and take p
,
q with 1
≤
p
≤
q
<
∞. A functional Λ
∈
L
∞
(
G
)
* is left (
p
,
q
)-multiinvariant if the set {
s
·
Λ
:
s
∈
G
} is multibounded in the (
p
,
q
)-multinorm. The group G is left (
p
,
q
)-amenable if there exists a left (
p
,
q
)-multiinvariant mean on L
∞
(
G
).The idea behind this definition is to attempt to measure the “left invariance” of a meanΛ
∈
L
∞
(
G
)
* by measuring the growth of the sets {
s
·
Λ
:
s
∈
F
} as F ranges through all the finite subsets of G. See [2] for more details.


## 2.1. Weighted Group Algebras and Approximate Amenability of Banach Algebras

LetG denote a locally compact group with a fixed left Haar measurable λ and ω a weight function on G; that is, a Borel measurable function ω
:
G
→
ℝ
+ such that
(1)
ω
(
x
·
y
)
≤
ω
(
x
)
ω
(
y
)
,
(
x
,
y
∈
G
)
.
The weighted group algebra L
1
(
G
,
ω
) is the space of all measurable complex-valued functions f on G such that
(2)
∫
G
‍
|
f
(
x
)
|
ω
(
x
)
d
x
<
∞
and is equipped with the convolution product * of functions; that is, for f
,
g
∈
L
1
(
G
,
ω
) and x
∈
G,
(3)
(
f
*
g
)
(
x
)
=
∫
G
‍
f
(
x
y
-
1
)
g
(
y
)
d
y
,
and the norm
(4)
∥
f
∥
ω
=
∫
G
‍
|
f
(
x
)
|
ω
(
x
)
d
x
.
Also, let L
∞
(
G
,
ω
-
1
) be the space of all measurable complex-valued functions ϕ on G, such that ϕ
/
ω is essentially bounded, and for ϕ
∈
L
∞
(
G
,
ω
-
1
) define
(5)
∥
ϕ
∥
∞
,
ω
=
∥
ϕ
ω
∥
∞
=
ess sup
{
|
ϕ
(
x
)
ω
(
x
)
|
∣
x
∈
G
}
.
The spaces L
1
(
G
,
ω
) and L
∞
(
G
,
ω
-
1
) are in duality by
(6)
〈
f
,
ϕ
〉
=
∫
G
‍
f
ϕ
d
λ
(
f
∈
L
1
(
G
,
ω
)
,
ϕ
∈
L
∞
(
G
,
ω
-
1
)
)
,
and if ω
(
x
)
≥
1, (
x
∈
G
), then L
∞
(
G
)
⊆
L
∞
(
G
,
ω
-
1
). Let M
(
G
,
ω
) be the Banach space of all complex-valued, regular Borel measures μ on G such that
(7)
∥
μ
∥
ω
=
∫
G
‍
ω
(
x
)
d
|
μ
|
(
x
)
<
∞
.
Note that M
(
G
,
1
)
=
M
(
G
). If ω
(
x
)
≥
1, (
x
∈
G
), then M
(
G
,
ω
) is a subalgebra of M
(
G
).Let𝒜 be a Banach algebra and X a Banach 𝒜-bimodule. A bounded linear map D
:
𝒜
→
X is called a derivation if
(8)
D
(
a
b
)
=
D
(
a
)
·
b
+
a
·
D
·
(
b
)
(
a
,
b
∈
𝒜
)
.
For every x
∈
X we define a
d
x
𝒜 by
(9)
a
d
x
𝒜
(
a
)
=
a
x
-
x
a
(
a
∈
𝒜
)
.
It is easily seen that a
d
x
𝒜 is a derivation. Derivation of this form is called inner derivation.Let𝒜 be a Banach algebra and X a Banach 𝒜-bimodule. Then X
*, the dual space of X, is a Banach 𝒜-bimodule for operations given by
(10)
〈
x
,
a
ξ
〉
=
〈
x
a
,
ξ
〉
,
〈
x
,
ξ
a
〉
=
〈
a
x
,
ξ
〉
111111111
(
a
∈
𝒜
,
x
∈
X
,
ξ
∈
X
*
)
.
X
* is the dual module of X; and in particular 𝒜
* is the dual module of 𝒜.A Banach algebra𝒜 is called amenable if, for any 𝒜-bimodule X, any derivation D
:
𝒜
→
X
* is inner. This definition of amenability was introduced by Johnson in [3]. A Banach algebra 𝒜 is called weakly amenable if any derivation D
:
𝒜
→
𝒜
* is inner. Trivially, any amenable Banach algebra is weakly amenable.LetG be a locally compact group; a mean on L
∞
(
G
) is a positive functional m
∈
L
∞
(
G
)
* such that
(11)
〈
1
,
m
〉
=
∥
m
∥
=
1
.
A mean m on L
∞
(
G
) is called left invariant if
(12)
〈
δ
x
*
ϕ
,
m
〉
=
〈
ϕ
,
m
〉
(
x
∈
G
,
ϕ
∈
L
∞
(
G
)
)
.
A locally compact group G is called amenable if there exists a left invariant mean on L
∞
(
G
). Note that G is amenable if and only if L
1
(
G
) is an amenable Banach algebra.A derivationD
:
𝒜
→
X is called approximately inner if there exists net (
ξ
a
)
⊆
X such that, for every a
∈
𝒜,
(13)
D
(
a
)
=
norm-
lim
⁡
a
⁡
(
a
·
ξ
a
-
ξ
a
·
a
)
.
Recall from [1] that a Banach algebra 𝒜 is called approximately amenable if, for any 𝒜-bimodule X, any derivation D
:
𝒜
→
X
* is approximately inner.


## 2.2. Banach Spaces

Forn
∈
ℕ
=
{
1,2
,
…
}, we set ℕ
n
=
{
1
,
…
,
n
}. Let E be a normed space. For each k
∈
ℕ, we denote by E
k the linear space direct product of k copies of E. The closed unit ball of E is denoted by B
1
(
E
). We denote the dual of E by E
*; the action of λ
∈
E
* on an element x
∈
E is written as 〈
x
,
λ
〉.Following the notation of [4, 5] we define the weak
p
-summing norm (for 1
≤
p
<
∞) on E
n by
(14)
μ
p
,
n
(
x
)
=
sup
⁡
{
(
∑
i
=
1
n
‍
|
〈
x
i
,
λ
〉
|
p
)
1
/
p
:
λ
∈
B
1
(
E
*
)
}
,
where x
=
(
x
1
,
…
,
x
n
)
∈
E
n. See also [6, 7]. Notice that, by the weak* density of B
1
(
E
) in B
1
(
E
*
*
), the weak p-summing norm on (
E
*
)
n can also be computed as
(15)
μ
p
,
n
(
λ
)
=
sup
⁡
{
(
∑
i
=
1
n
‍
|
〈
x
,
λ
i
〉
|
p
)
1
/
p
:
x
∈
B
1
(
E
)
}
,
where λ
=
(
λ
1
,
…
,
λ
n
)
∈
(
E
*
)
n.


## 2.3. Multinormed Spaces

The following definition is by Dales and Polyakov. For a full account of the theory of multinormed spaces, see [4].Definition 1.
Let(
E
,
∥
·
∥
) be a normed space, and let (
∥
·
∥
n
:
n
∈
ℕ
) be a sequence such that ∥
·
∥
n is a norm on E
n for each n
∈
ℕ, with ∥
·
∥
1
=
∥
·
∥ on E. Then the sequence (
∥
·
∥
n
:
n
∈
ℕ
) is a multinorm if the following axioms hold (where in each case the axiom is required to hold for all n
≥
2 and all x
1
,
…
,
x
n
∈
E): (A1)
∥
x
σ
(
1
)
,
…
,
x
σ
(
n
)
∥
n
=
∥
(
x
1
,
…
,
x
n
)
∥
n for each permutation σ of ℕ
n;
(A2)
∥
(
α
1
x
1
,
…
,
α
n
x
n
)
∥
n
≤
max
⁡
i
∈
ℕ
n
|
α
i
|
∥
(
x
1
,
…
,
x
n
)
∥
n  (
α
1
,
…
,
α
n
)
∈
ℂ;
(A3)
∥
(
x
1
,
…
,
x
n
-
1
,
0
)
∥
n
=
∥
(
x
1
,
…
,
x
n
-
1
)
∥
n
-
1;
(A4)
∥
(
x
1
,
…
,
x
n
-
2
,
x
n
-
1
)
∥
n
=
∥
(
x
1
,
…
,
x
n
-
2
,
x
n
-
1
)
∥
n
-
1.
The normed spaceE equipped with a multinorm is a multinormed space, denoted in full by (
(
E
n
,
∥
·
∥
n
)
:
n
∈
ℕ
). We say that such a multinorm is based on E.Definition 2.
Let(
(
E
n
,
∥
·
∥
n
:
n
∈
ℕ
) be a multinormed space. A subset B
⊂
E is multibounded if
(16)
mb
(
B
)
:
=
sup
⁡
{
∥
(
x
1
,
…
,
x
n
)
∥
n
:
x
1
,
…
,
x
n
∈
B
,
n
∈
ℕ
}
<
∞
.
The constant mb
(
B
) is the multibound of B.The following easy remark is from [4, Proposition 6.5
(
ii
)].Remark 3.
LetE be a multinormed space. Then the absolutely convex hull of a multibounded set is multibounded, with the same multibound.


## 2.4. The(
p
,
q
)-Multinorm

Following [4], we now introduce an important class of multinorms. Let E be a normed space, and take p
,
q with 1
≤
p, q
<
∞. For each n
∈
ℕ and each x
=
(
x
1
,
…
,
x
n
)
∈
E
n, we define
(17)
∥
x
∥
n
(
p
,
q
)
=
sup
⁡
{
(
∑
i
=
1
n
|
〈
x
i
,
λ
i
〉
|
q
‍
)
1
/
q
:
11111111
λ
=
(
λ
1
,
…
,
λ
n
)
∈
(
E
*
)
n
,
μ
p
,
n
(
λ
)
≤
1
(
∑
i
=
1
n
|
〈
x
i
,
λ
i
〉
|
q
‍
)
1
/
q
}
.
It is clear that ∥
·
∥
n
(
p
,
q
) is a norm on E
n. As proved in [4], in the case where 1
≤
p
≤
q
<
∞, the sequence (
∥
·
∥
n
(
p
,
q
)
:
n
∈
ℕ
) is a multinorm based on E.Definition 4.
LetE be a normed space, and take p
,
q with 1
≤
p
≤
q
<
∞. Then the multinorm (
∥
·
∥
n
(
p
,
q
)
:
n
∈
ℕ
) described above is the (
p
,
q
)-multinorm over E.
A subset ofE is (
p
,
q
)-multibounded if it is multibounded with respect to the (
p
,
q
)-multinorm. The (
p
,
q
)-multibound of such a set B is denoted by m
b
p
,
q
(
B
).Lemma 5.
LetE be a normed space, and take p
,
q with 1
≤
p, q
<
∞. Then, for each n
∈
ℕ and λ
=
(
λ
1
,
…
,
λ
n
)
∈
(
E
*
)
n, we have
(18)
∥
λ
∥
n
(
p
,
q
)
=
sup
⁡
{
(
∑
i
=
1
n
‍
|
〈
x
i
,
λ
i
〉
|
q
)
1
/
q
:
hhhhhhhh
x
=
(
x
1
,
…
,
x
n
)
∈
(
E
*
)
n
,
μ
p
,
n
(
x
)
≤
1
(
∑
i
=
1
n
‍
|
〈
x
i
,
λ
i
〉
|
q
)
1
/
q
}
.Proof.
This is proved in [4, Proposition 4.10]; it follows from the Principal of Local Reflexivity.


## 2.5. Left(
p
,
q
)-Multiinvariant Means

Definition 6.
LetG be a locally compact group, and take p
,
q with 1
≤
p
≤
q
<
∞. A functional Λ
∈
L
∞
(
G
)
* is left (
p
,
q
)-multiinvariant if the set {
s
·
Λ
:
s
∈
G
} is multibounded in the (
p
,
q
)-multinorm. The group G is left (
p
,
q
)-amenable if there exists a left (
p
,
q
)-multiinvariant mean on L
∞
(
G
).The idea behind this definition is to attempt to measure the “left invariance” of a meanΛ
∈
L
∞
(
G
)
* by measuring the growth of the sets {
s
·
Λ
:
s
∈
F
} as F ranges through all the finite subsets of G. See [2] for more details.


## 3. Multinorms and Approximate Amenability of Weighted Group Algebras

Ghahramani and Loy in [1] proved that for a locally compact group G the Banach algebra L
1
(
G
) is approximately amenable if and only if G is amenable. In this section we prove that, if ω
≥
1 is any weight function on G and L
1
(
G
,
ω
) is approximately amenable, then G is left (
p
,
q
)-amenable. Through an example we show is not valid in general.Theorem 7.
LetG be a locally compact group, and let ω be a weight function on G such that ω
(
x
)
≥
1  (
x
∈
G
). If L
1
(
G
,
ω
) is approximately amenable, then G is left (
p
,
q
)-amenable.Proof.
By the following operations,L
∞
(
G
,
ω
-
1
) is a Banach M
(
G
,
ω
)-bimodule;
(19)
〈
f
,
ϕ
·
μ
〉
=
〈
μ
*
f
,
ϕ
〉
,
μ
·
ϕ
=
μ
(
G
)
ϕ
,
where μ
∈
M
(
G
,
ω
), ϕ
∈
L
∞
(
G
,
ω
-
1
), and f
∈
L
1
(
G
,
ω
). Note that we have
(20)
(
μ
*
f
)
(
x
)
=
∫
G
‍
f
(
y
-
1
x
)
d
μ
(
y
)
for μ
∈
M
(
G
,
ω
), f
∈
L
1
(
G
,
ω
), and x
∈
G. For every μ
∈
M
(
G
,
ω
) and f
∈
L
∞
(
G
,
ω
-
1
),
(21)
〈
f
,
1
·
μ
〉
=
〈
μ
*
f
,
1
〉
=
∫
‍
(
μ
*
f
)
(
x
)
d
x
=
∬
f
(
y
-
1
x
)
d
μ
(
y
)
d
x
=
∬
f
(
x
)
d
x
d
μ
(
y
)
=
μ
(
G
)
∫
‍
f
(
x
)
d
x
=
μ
(
G
)
〈
f
,
1
〉
.Thus1
·
μ
=
μ
(
G
)
·
1
∈
ℂ
1; and by definition we have μ
·
1
∈
ℂ
1; that is, the space ℂ
1 is a submodule of M
(
G
,
ω
). If we set Z
=
L
∞
(
G
,
ω
-
1
)
/
ℂ
1, then Z is a M
(
G
,
ω
)-bimodule, with Z
*
=
{
m
∈
L
∞
(
G
,
ω
-
1
)
*
∣
m
(
1
)
=
0
}. By Hahn Banach Theorem there is v
∈
L
∞
(
G
,
ω
-
1
)
* such that v
(
1
)
=
1. It is easy to prove that the mapping D
~
:
M
(
G
,
ω
)
→
Z
*, defined by
(22)
D
~
(
μ
)
(
ϕ
+
ℂ
1
)
=
(
μ
·
v
-
μ
(
G
)
v
)
(
ϕ
)
(
μ
∈
M
(
G
,
ω
)
,
ϕ
∈
L
∞
(
G
,
ω
-
1
)
)
,
is a well-defined derivation. If the restrict ion D
~ to L
1
(
G
,
ω
) is denoted by D, then from the approximate amenability of L
1
(
G
,
ω
) it follows that there is a net (
m
i
) in Z
* such that for every μ
∈
L
1
(
G
,
ω
) we have
(23)
D
(
μ
)
=
μ
·
v
-
v
μ
(
G
)
=
lim
⁡
i
(
μ
·
m
i
-
m
i
μ
(
G
)
)
.
For μ
∈
L
1
(
G
,
ω
) with μ
(
G
)
=
1 and ϕ
∈
L
∞
(
G
,
ω
-
1
),  μ
ϕ
=
ϕ. Hence for every x
∈
G, ϕ
∈
L
∞
(
G
,
ω
-
1
), we have
(24)
〈
ϕ
+
ℂ
1
,
D
(
δ
x
)
·
μ
〉
=
〈
μ
(
ϕ
+
ℂ
1
)
,
D
(
δ
x
)
〉
=
〈
μ
·
ϕ
+
ℂ
1
,
D
(
δ
x
)
〉
=
〈
ϕ
+
ℂ
1
,
D
(
δ
x
)
〉
.
Thus D
(
δ
x
)
·
μ
=
D
(
δ
x
). Therefore for every x
∈
G and μ
∈
L
1
(
G
,
ω
) with μ
(
G
)
=
1 we have
(25)
δ
x
·
v
-
v
=
D
(
δ
x
)
=
D
(
δ
x
)
·
μ
=
D
(
δ
x
*
μ
)
-
δ
x
D
(
μ
)
=
lim
⁡
i
⁡
[
(
δ
x
*
μ
)
·
m
i
-
m
i
(
δ
x
*
μ
)
11111
-
δ
x
(
μ
·
m
i
-
m
i
·
μ
)
]
=
lim
⁡
i
[
(
δ
x
*
μ
)
·
m
i
11111
-
m
i
-
δ
x
(
μ
·
m
i
-
m
i
)
]
=
lim
⁡
i
(
δ
x
m
i
-
m
i
)
.
Thus
(26)
lim
⁡
i
δ
x
(
v
-
m
i
)
-
(
v
-
m
i
)
=
0
.
Since 〈
v
-
m
i
,
1
〉
=
1, it follows that lim
⁡
i
∥
v
-
m
i
∥
≠
0, so there exists a subnet (
m
j
) of (
m
i
) such that ∥
v
-
m
~
j
∥
≠
0. For every j, if we set
(27)
k
~
j
=
v
-
m
~
j
∥
v
-
m
~
j
∥
,
then k
~
j
∈
L
∞
(
G
,
ω
-
1
). If for each j the restriction of k
~
j to L
∞
(
G
) is denoted by (26) we have norm-lim
⁡
j
(
δ
x
·
k
j
-
k
j
)
=
0, for every x
∈
G. Since the space L
∞
(
G
) is a commutative C
*-algebra with identity 1, there is a compact space T such that L
∞
(
G
)
*
=
M
(
T
). Since, for each x
∈
G and m
∈
M
(
T
), we have |
δ
x
·
m
|
=
δ
x
|
m
|, therefore
(28)
|
δ
x
·
|
k
j
|
-
|
k
j
|
|
=
|
|
δ
x
·
k
j
|
-
|
k
j
|
|
≤
|
δ
x
·
k
j
-
k
j
|
(
x
∈
G
)
,
and since lim
⁡
j
|
δ
x
·
k
j
-
k
j
|
=
0, thus, for each x
∈
G,
(29)
norm-
lim
⁡
j
(
δ
x
·
|
k
j
|
-
|
k
j
|
)
=
0
.
Let k be a weak*-cluster point of (
k
j
). Clearly k is positive; take p, q with 1
≤
p
≤
q
<
∞. Recall that L
∞
(
G
)
* can be identified isometrically as a Banach lattice with L
1
(
Ω
) for some measure space (
Ω
,
μ
). Set k
^
=
|
k
|
/
∥
k
∥. Since ∥
|
k
|
∥
=
∥
k
∥
=
〈
1
,
|
k
|
〉, it is clear that k
^ is a mean on L
∞
(
G
). Since μ
p
,
n
(
φ
1
,
…
,
φ
n
)
=
μ
p
,
n
(
ψ
1
,
…
,
ψ
n
) for every n
∈
ℕ and every φ
1
,
…
,
φ
n
,
ψ
1
,
…
,
ψ
n
∈
L
∞
(
Ω
) with |
φ
i
|
=
|
ψ
i
|  (
i
∈
ℕ
n
) [5, 2.6], we see that
(30)
∥
(
k
1
,
…
,
k
n
)
∥
n
(
p
,
q
)
=
∥
(
|
k
1
|
,
…
,
|
k
n
|
)
∥
n
(
p
,
q
)
11111111111111
(
k
1
,
…
,
k
n
∈
L
∞
(
G
)
*
)
.
Now note that |
s
·
k
|
=
s
·
|
k
| for every s
∈
G, and so {
s
·
k
^
:
s
∈
G
} is multibounded in the (
p
,
q
)-multinorm. The result follows, and G is left (
p
,
q
)-amenable.Lemma 8.
Let𝒜 be a commutative Banach algebra. If 𝒜 is approximately amenable, then 𝒜 is weakly amenable.Proof.
Suppose thatD
:
𝒜
→
𝒜
* is a derivation we show that D is inner. Since 𝒜 is approximately amenable and commutative, so there exists a net (
ξ
i
) in 𝒜
* such that for each a
∈
𝒜
(31)
D
(
a
)
=
norm-
lim
⁡
i
(
a
·
ξ
i
-
ξ
i
·
a
)
=
0
.
On the other hand since 𝒜 is commutative, every inner derivation on 𝒜 is zero, so D is inner. Therefore 𝒜 is weakly amenable.The Example 6.2 of [1] shows that an approximately amenable Banach algebra need not be weakly amenable.The following example shows that the opposite of Theorem7 is not true in general.Example 9.
If we define the weight functionω on (
ℤ
,
+
) by ω
(
n
)
=
1
+
|
n
| for every n
∈
ℤ, then the Banach algebra l
1
(
ℤ
,
ω
) is not approximately amenable.Proof.
Let(32)
l
1
(
ℤ
,
ω
)
=
{
a
=
(
a
(
n
)
:
n
∈
ℤ
)
∣
∑
-
∞
∞
‍
|
a
(
n
)
|
ω
(
n
)
≤
∞
}
.
Then l
1
(
ℤ
,
ω
) is a commutative Banach algebra with respect to convolution multiplication,
(33)
(
a
*
b
)
(
n
)
=
∑
k
=
-
∞
∞
‍
a
(
n
-
k
)
b
(
k
)
1111111
(
n
∈
ℤ
,
a
,
b
∈
l
1
(
ℤ
,
ω
)
)
,
and the norm
(34)
∥
a
∥
=
∑
-
∞
∞
‍
|
a
(
n
)
|
ω
(
n
)
<
∞
(
a
∈
l
1
(
ℤ
,
ω
)
)
.
We show that l
1
(
ℤ
,
ω
) is not approximately amenable. To see this note that
(35)
sup
⁡
{
ω
(
n
+
m
)
ω
(
n
)
ω
(
m
)
·
1
+
|
n
|
1
+
|
n
+
m
|
∣
n
,
m
∈
ℤ
}
hhhh
=
sup
⁡
{
1
1
+
|
m
|
∣
m
∈
ℤ
}
=
1
.
So by Theorem 2.3 of [8], l
1
(
ℤ
,
ω
) is not weakly amenable. On the other hand, since l
1
(
ℤ
,
ω
) is a commutative Banach algebra and if it is approximately amenable, so by Lemma 8 it is weakly amenable, and this is a contradiction. So l
1
(
ℤ
,
ω
) is not approximately amenable.Letω be a weight function on G; for x
∈
G, we define ω
*
(
x
)
=
ω
(
x
)
ω
(
x
-
1
). In the following theorem we prove that if ω
* is bounded on G then the opposite of Theorem 7 is true.Theorem 10.
Suppose thatω is a weight function on a locally compact group G such that ω
≥
1 and ω
* is bounded. Then G is left (
p
,
q
)-amenable if and only if L
1
(
G
,
ω
) is approximately amenable.Proof.
G is left (
p
,
q
)-amenable, so suppose that Λ is a left (
p
,
q
)-invariant mean on L
∞
(
G
); that is, {
s
·
Λ
:
s
∈
G
} is (
p
,
q
)-multibounded on L
∞
(
G
)
*. By Remark 3 or the Krein-Smulian theorem, the closed convex hull K of {
s
·
Λ
:
s
∈
G
} is weakly compact. For each s
∈
G, consider that the map L
s
:
K
→
K defined by
(36)
L
s
(
φ
)
=
s
·
φ
(
s
∈
G
,
φ
∈
K
)
.
We obtain a group Σ
:
=
{
L
s
:
s
∈
G
} of isometric affine maps. By Ryll-Nardzewski fixed point theorem, given in [9, 10], there exists Λ
0
∈
K which is a common fixed point for the set {
L
s
:
s
∈
G
}. Obviously, Λ
0 must be a left-invariant mean on L
∞
(
G
). Hence the group G is amenable and ω
* is bounded on G; by Theorem 0 of [11] L
1
(
G
,
ω
) is amenable, so L
1
(
G
,
ω
) is approximately amenable.


## 4. Approximate AmenabilityM
(
G
,
ω
) of a Locally Compact Group G

It is standard thatL
1
(
G
) always has a bounded approximate identity which is a net consisting of continuous functions of compact support, and this net is clearly also a bounded approximate identity for L
1
(
G
,
ω
).Letω be a weight function on G with ω
(
x
)
≥
1 (x
∈
G); then, with the convolution product * given by
(37)
〈
ϕ
,
μ
*
ν
〉
=
∬
G
ϕ
(
s
t
)
d
μ
(
s
)
d
ν
(
t
)
(
μ
,
ν
∈
M
(
G
,
Ω
)
,
ϕ
∈
C
0
(
G
,
ω
)
)
,
the Banach space M
(
G
,
ω
) defines a unital convolution Banach algebra for which L
1
(
G
,
ω
) is a closed ideal.In the following lemma we show that ifM
(
G
,
ω
) is approximately amenable then G is left (
p
,
q
)-amenable.Theorem 11.
LetG be a locally compact group. If M
(
G
,
ω
) is approximately amenable then G is left (
p
,
q
)-amenable.Proof.
SinceL
1
(
G
,
ω
) is a closed two sided ideal of M
(
G
,
ω
) and has an approximate identity, from Corollary 2.3 of [1] it follows that L
1
(
G
,
ω
) is approximately amenable. So by Theorem 7, G is left (
p
,
q
)-amenable.It is well known (c.f. [12]) that if G is amenable and ω
* is bounded on G, then there is a continuous positive character ϕ on G such that
(38)
ϕ
≤
ω
≤
c
ϕ
such
that
c
=
sup
⁡
{
ω
*
(
x
)
∣
x
∈
G
}
.
In particular,
(39)
L
1
(
G
,
ω
)
=
L
1
(
G
,
ϕ
)
≃
L
1
(
G
)
,
M
(
G
,
ω
)
=
M
(
G
,
ϕ
)
≃
M
(
G
)
.Theorem 12.
IfM
(
G
,
ω
) is approximately amenable and ω
* is bounded on G, then G is a discrete group.Proof.
SinceM
(
G
,
ω
) is approximately amenable, from Theorem 11 we conclude that G is left (
p
,
q
)-amenable. Using the fact that ω
* is bounded, we infer that M
(
G
,
ω
)
≃
M
(
G
). So by Theorem 1.1 of [13] G is discrete.



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*Source: 410262-2014-02-23.xml*