# Amenability and Contractibility Modulo an Ideal of Banach Algebras

Authors: Hamidreza Rahimi; Elham Tahmasebi
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We investigate the concept of amenability modulo an ideal of Banach algebra, showing that amenability modulo an ideal can be characterized by the existence of virtual and approximate diagonal modulo an ideal. We also study the concept of contractible modulo an ideal of Banach algebra. As a consequence, we prove a version of Selivanov's theorem for a large class of semigroups, includingE-inversive E-semigroup and eventually inverse semigroups.


---

## Body

## 1. Introduction

Introduced and initiated in 1972 by Johnson [1] for groups, the concept of amenability of Banach algebras has since become an important group-theoretic and semigroup-theoretic tool, and since then many results have been obtained on the structure of this concept. This notion was coined by Johnson [1] in recognition of “a locally compact (discrete) group G is amenable (in the usual sense) if and only if L
1
(
G
) (resp., l
1
(
G
)) is amenable.’’ For semigroups, in especial cases some necessary and sufficient conditions for amenability of semigroup algebra were introduced (see, e.g., [2–5] for more details).Recently in [6], the first author and Amini introduced and studied the concept of amenability modulo an ideal. They showed that inducing the amenability of l
1
(
S
) modulo ideals by certain classes of group congruences on S is equivalent to the amenability of S, restoring Johnson's theorem for a large class of semigroups.In this paper, we study basic properties of amenability modulo an ideal such as virtual diagonal modulo an ideal, approximate diagonal modulo an ideal, and contractible modulo an ideal. Among other results, we show that the semigroup algebral
1
(
S
) is contractible modulo an ideal if and only if S
/
σ is finite, restoring Selivanov's theorem for a large class of semigroups [7].The paper is organized as follows. In Section2 we introduce the notions of virtual and approximate diagonal modulo an ideal. As an application, we characterize amenability modulo an ideal of Banach algebra. In Section 3 we consider the concept of contractible modulo an ideal of Banach algebra. In Section 4, constructing various examples we investigate the contractibility modulo an ideal of semigroup algebra.


## 2. Virtual and Approximate Diagonal Modulo an Ideal

LetA be a Banach algebra and let X be a Banach A-bimodule. By a derivation of A to X we mean a bounded linear map D
:
A
→
X such that D
(
a
b
)
=
a
·
D
(
b
)
+
D
(
a
)
·
b
(
a
,
b
∈
A
). Clearly the mapping a
d
x
:
A
→
X defined by a
d
x
(
a
)
=
a
·
x
-
x
·
a is a derivation that is called inner (x
∈
X). A Banach algebra A is called amenable if, for any A-bimodule X and any derivation D
:
A
→
X
*, there exists ϕ
∈
X
* such that D
=
a
d
ϕ.In the following we recall the concept of amenability modulo an ideal and some results of [6] which we need in the future.Definition 1.
LetI be a closed ideal of A. A Banach algebra A is amenable modulo I if for every Banach A-bimodule E such that I
·
E
=
E
·
I
=
0 and every derivation D from A into E
* there is ϕ
∈
E
* such that D
=
a
d
ϕ on the set theoretic difference A
∖
I
≔
{
a
∈
A
:
a
∉
I
}.Proposition 2 (see [6, Theorem 1]).
LetI be a closed ideal of A.(i)
IfA
/
I is amenable and I
2
=
I, then A is amenable modulo I.
(ii)
IfA is amenable modulo I, then A
/
I is amenable.
(iii)
IfA is amenable modulo I and I is amenable, then A is amenable.Blanket Assumption. All over this paper we fix A and I as above, unless they are otherwise specified.Definition 3.
LetA be a Banach algebra. A bounded net {
u
α
}
α
⊆
A is approximate identity modulo I if
(1)
lim
⁡
α
u
α
·
a
=
lim
⁡
α
a
·
u
α
=
a
(
a
∈
A
∖
I
)
.Theorem 4.
LetA be amenable modulo I. Then A has an approximate identity modulo I.Proof.
SetX
=
A
/
I; then X is a Banach A-bimodule where it acts on A via
(2)
a
·
π
(
b
)
≔
π
(
a
)
π
(
b
)
,
π
(
b
)
·
a
≔
0
(
a
,
b
∈
A
)
.
Clearly I
·
X
*
=
X
*
·
I
=
0. Suppose D
=
ι
∘
π
:
A
→
X
*
*, where ι
:
X
→
X
*
* is a natural inclusion map and π
:
A
→
X is a natural quotient map. Since D is a derivation, there exists ϕ
∈
X
*
* such that D
(
a
)
=
a
·
ϕ
(
a
∈
A
∖
I
). Goldstine's theorem provides a bounded net {
u
α
}
α in A such that ϕ
=
w
*
-
lim
⁡
α
u
~
α. Then ϕ
=
w
*
-
lim
⁡
α
D
(
u
α
). Now
(3)
w
*
-
lim
⁡
α
D
(
a
·
u
α
)
=
w
*
-
lim
⁡
α
(
D
(
a
)
·
u
α
+
a
·
D
(
u
α
)
)
=
a
·
w
*
-
lim
⁡
α
D
(
u
α
)
=
a
·
ϕ
=
D
(
a
)
(
a
∈
A
∖
I
)
which is equivalent to w
-
lim
⁡
α
a
·
u
α
=
a
(
a
∈
A
∖
I
). Passing to convex combinations, we can suppose that a
=
lim
⁡
α
a
·
u
α
(
a
∈
A
∖
I
). In an analogous fashion, we obtain a net {
v
β
}
β in A such that
(4)
a
=
w
-
lim
⁡
β
v
β
·
a
(
a
∈
A
∖
I
)
.
Define e
α
,
β
≔
u
α
+
v
β
-
u
α
v
β. Then e
α
,
β is an approximate identity modulo I.We now express the version of Cohen's factorization theorem [8], where A has an approximate identity modulo I. Since the proof is similar, we omit the proof.Theorem 5.
IfA has a bounded approximate identity modulo I, z
∈
A
∖
I, and δ
>
0, then there exist a
,
y
∈
A such that z
=
a
y and ∥
z
-
y
∥
≤
δ.LetA be a Banach algebra and let I be a closed ideal of A; then the corresponding diagonal operator of A
/
I is defined as
(5)
π
A
/
I
:
A
I
⊗
^
A
I
⟶
A
I
,
(
a
+
I
)
⊗
(
b
+
I
)
⟶
a
b
+
I
.
Clearly, π
A
/
I is a bimodule homomorphism.Definition 6.
(i) An elementM
∈
(
(
A
/
I
)
⊗
^
(
A
/
I
)
)
*
* is a virtual diagonal modulo I if
(6)
a
·
π
A
/
I
*
*
M
-
a
~
=
0
(
a
∈
A
,
a
~
=
a
+
I
)
,
a
·
M
-
M
·
a
=
0
(
a
∈
A
∖
I
)
.
(ii) A bounded net(
m
α
)
α
⊂
(
A
/
I
)
⊗
^
(
A
/
I
) is an approximate diagonal modulo I if
(7)
a
·
π
A
/
I
m
α
-
a
~
⟶
0
(
a
∈
A
,
a
~
=
a
+
I
)
,
a
·
m
α
-
m
α
·
a
⟶
0
(
a
∈
A
∖
I
)
.
(iii) An elementm
∈
(
A
/
I
)
⊗
^
(
A
/
I
) is a diagonal modulo I if
(8)
a
·
π
A
/
I
m
-
a
~
=
0
(
a
∈
A
,
a
~
=
a
+
I
)
,
a
·
m
-
m
·
a
=
0
(
a
∈
A
∖
I
)
.Theorem 7.
The following conditions are equivalent:(i)
A is amenable modulo I,
(ii)
there is an approximate diagonal moduloI,
(iii)
there is a virtual diagonal moduloI.Proof.
(i) ⇒ (iii) SinceA is amenable modulo I, A
/
I is amenable. Thus A
/
I has a bounded approximate identity (
e
α
)
α. Let E
∈
(
(
A
/
I
)
⊗
^
(
A
/
I
)
)
*
* be a w
*-accumulation point of (
e
α
⊗
e
α
)
α. Put a
d
E
:
A
→
(
(
A
/
I
)
⊗
^
(
A
/
I
)
)
*
*
,
a
↦
a
·
E
-
E
·
a; then
(9)
π
A
/
I
*
*
(
a
·
E
-
E
·
a
)
=
w
*
-
lim
⁡
α
π
A
/
I
(
a
·
(
e
α
⊗
e
α
)
-
(
e
α
⊗
e
α
)
·
a
)
=
lim
⁡
α
a
e
α
2
-
e
α
2
a
=
0
(
∀
a
∈
A
)
.
Theorem 5 implies that the map π
A
/
I is surjective; then (
ker
⁡
π
A
/
I
)
*
*
=
ker
⁡
π
A
/
I
*
*. Furthermore (
A
/
I
)
⊗
^
(
A
/
I
) can be made into Banach A-bimodule by defining
(10)
a
·
e
=
(
a
+
I
)
·
e
,
e
·
a
=
e
·
(
a
+
I
)
(
a
∈
A
,
e
∈
A
I
⊗
^
A
I
)
.
Since π
A
/
I is a closed submodule of (
(
A
/
I
)
⊗
^
(
A
/
I
)
), ker
⁡
π
A
/
I
*
* is a dual Banach A-bimodule. Let X
≔
(
(
A
/
I
)
⊗
^
(
A
/
I
)
)
* and M
≔
(
π
A
/
I
*
(
A
/
I
)
*
)
¯; then ker
⁡
π
A
/
I
*
*
=
M
⊥
=
(
X
/
M
)
*. Since I
·
(
X
/
M
)
=
(
X
/
M
)
·
I
=
0, amenability modulo I of A implies that there exists F
∈
ker
⁡
π
A
/
I
*
* such that a
d
E
=
a
d
F on A
∖
I. Define M
′
≔
E
-
F; then
(11)
a
·
π
A
/
I
*
*
(
M
′
)
=
a
·
π
A
/
I
*
*
(
E
-
F
)
=
a
·
π
A
/
I
*
*
(
E
)
-
a
·
π
A
/
I
*
*
(
F
)
=
lim
⁡
α
a
e
α
2
-
0
=
a
+
I
(
a
∈
A
)
,
a
·
M
′
-
M
′
·
a
=
a
·
(
E
-
F
)
-
(
E
-
F
)
·
a
=
a
d
E
(
a
)
-
a
d
F
(
a
)
=
0
(
a
∈
A
∖
I
)
.
(ii) ⇒ (i) Letm
α
=
∑
n
=
1
∞
‍
a
n
(
α
)
⊗
b
n
(
α
), where a
n
(
α
)
=
c
n
(
α
)
+
I and b
n
(
α
)
=
d
n
(
α
)
+
I
(
α
∈
I
,
n
∈
N
) with ∑
n
=
1
∞
‍
∥
c
n
(
α
)
∥
∥
d
n
(
α
)
∥
<
∞. Suppose E is a Banach A-bimodule such that E
·
I
=
I
·
E
=
0 and D
:
A
→
E
* is a bounded derivation. Then (
∑
n
=
1
∞
‍
c
n
(
α
)
·
D
d
n
(
α
)
)
α is a bounded net in E
*, which has a w
*-accumulation point; say ϕ
∈
E
*. Let F
:
A
→
E
* by F
(
a
)
=
a
·
ϕ
-
ϕ
·
a
(
a
∈
A
). Clearly, F is an inner derivation and F
(
a
)
=
0
(
a
∈
I
). Let a
∈
A
∖
I and x
∈
E; then
(12)
〈
x
,
a
·
ϕ
〉
=
lim
⁡
α
〈
x
,
∑
n
=
1
∞
‍
a
c
n
(
α
)
·
D
(
d
n
(
α
)
)
〉
=
lim
⁡
α
〈
x
,
∑
n
=
1
∞
‍
c
n
(
α
)
·
D
(
d
n
(
α
)
·
a
)
〉
=
lim
⁡
α
〈
x
,
∑
n
=
1
∞
‍
[
c
n
(
α
)
d
n
(
α
)
·
D
(
a
)
w
w
w
w
w
w
w
w
i
+
c
n
(
α
)
D
(
d
n
(
α
)
)
·
a
]
〈
x
,
∑
n
=
1
∞
‍
[
c
n
(
α
)
d
n
(
α
)
·
D
(
a
)
〉
=
lim
⁡
α
〈
x
,
∑
n
=
1
∞
‍
c
n
(
α
)
d
n
(
α
)
·
D
(
a
)
〉
+
〈
x
,
ϕ
·
a
〉
=
〈
x
,
D
a
〉
+
〈
x
,
ϕ
·
a
〉
.
So a
·
ϕ
=
D
a
+
ϕ
·
a and D
=
a
d
ϕ on A
∖
I.
(iii) ⇒ (ii) LetM be a virtual diagonal modulo I and let (
m
α
)
α be a bounded net in (
A
/
I
)
⊗
^
(
A
/
I
) with M
=
w
*
-
lim
⁡
α
m
α. Then
(13)
w
-
lim
⁡
α
a
·
π
A
/
I
(
m
α
)
=
a
+
I
(
a
∈
A
)
,
w
-
lim
⁡
α
(
a
·
m
α
-
m
α
·
a
)
=
0
(
a
∈
A
∖
I
)
.
By passing to convex combinations, we obtain an approximate diagonal.Theorem 8.
LetA be amenable modulo I, let B be a Banach algebra, and let J be a closed ideal of B. Let θ
:
A
→
B be a continuous homomorphism with dens range such that θ
(
I
)
⊆
J. Then B is amenable modulo J.Proof.
SupposeE is a Banach B-bimodule such that J
·
E
=
E
·
J
=
0 and D
:
B
→
E
* is a bounded derivation. Then E becomes a Banach A-bimodule via
(14)
a
·
x
≔
θ
(
a
)
·
x
,
x
·
a
≔
x
·
θ
(
a
)
(
a
∈
A
,
x
∈
E
)
.
Clearly, I
·
E
=
E
·
I
=
0 and D
∘
θ
:
A
→
E
* is a bounded derivation. Since A is amenable modulo I, there exists ϕ
∈
E
* such that D
∘
θ
=
a
d
ϕ on A
∖
I. Let b
∈
B
∖
J; then there is a net (
a
α
)
α
⊂
A such that b
=
lim
⁡
α
θ
(
a
α
). Since θ
(
I
)
⊆
J, we may assume that (
a
α
)
α
⊂
A
∖
I. We have
(15)
D
(
b
)
=
lim
⁡
α
(
D
∘
θ
)
(
a
α
)
=
lim
⁡
α
a
α
·
ϕ
-
ϕ
·
a
α
=
lim
⁡
α
θ
(
a
α
)
·
ϕ
-
ϕ
·
θ
(
a
α
)
=
b
·
ϕ
-
ϕ
·
b
=
a
d
ϕ
(
b
)
.
Thus B is amenable modulo J.


## 3. Contractibility Modulo an Ideal of Banach Algebra

We recall that a Banach algebraA is contractible (super-amenable) if H
1
(
A
,
X
)
=
0 for every Banach A-bimodule X, where the left hand side is the first cohomology group of A with coefficient in X (see [3, 6]).Definition 9.
A Banach algebraA is contractible modulo I if for every Banach A-bimodule X such that I
·
X
=
X
·
I
=
0, every bounded derivation D from A into X is an inner derivation on the set theoretic difference A
∖
I
≔
{
a
∈
A
:
a
∉
I
}.Clearly, ifA is contractible modulo I, then A is amenable modulo I.Definition 10.
An elemente
∈
A is an identity modulo I if a
e
-
a
∈
I
,
e
a
-
a
∈
I
(
a
∈
A
). A Banach algebra A is unital modulo I if A has an identity modulo I.Lemma 11.
A has an identity modulo I if and only if A
/
I has an identity.Proof.
The elemente is an identity modulo I of A if and only if a
e
-
a
∈
I
,
e
a
-
a
∈
I. These relations are equivalent to (
a
+
I
)
(
e
+
I
)
=
a
+
I
,
(
e
+
I
)
(
a
+
I
)
=
a
+
I or e
+
I is an identity of A
/
I.Theorem 12.
The following assertions hold.(i)
IfA
/
I is contractible and I
2
=
I, then A is contractible modulo I.
(ii)
IfA is contractible modulo I, then A
/
I is contractible.
(iii)
If A is contractible moduloI and I is contractible, then A is contractible.
(iv)
IfA is contractible modulo I, then A
/
I is unital.
(v)
IfA is contractible modulo I, then A is unital modulo I.Proof.
(i) LetX be a Banach A-bimodule such that X
·
I
=
I
·
X
=
0 and let D
:
A
→
X be a bounded derivation. Since I
·
X
=
X
·
I
=
0, X with module action (
a
+
I
)
·
x
=
a
·
x can be regarded as A
/
I-bimodule. Let D
1
:
A
/
I
→
X by D
1
(
a
+
I
)
=
D
(
a
)
(
a
∈
A
). Since I
2
=
I, D
(
a
)
=
0
(
a
∈
I
). Hence D
1 is well-defined bounded derivation. Since A
/
I is contractible, there exits ϕ
∈
X such that D
1
=
a
d
ϕ. We have
(16)
D
(
a
)
=
D
1
(
a
+
I
)
=
(
a
+
I
)
·
ϕ
-
ϕ
·
(
a
+
I
)
=
a
·
ϕ
-
ϕ
·
a
=
a
d
ϕ
(
a
)
(
a
∈
A
∖
I
)
.
(ii) LetX be a Banach A
/
I-bimodule and let D
:
A
/
I
→
X be a bounded derivation. Now X can be made into Banach A-bimodule by defining
(17)
a
·
x
=
π
(
a
)
·
x
,
x
·
a
=
x
·
π
(
a
)
(
a
∈
A
,
x
∈
X
)
,
where π
:
A
→
A
/
I is the natural quotient map. Define D
∘
π
:
A
→
X. Clearly, D
∘
π is bounded derivation. Since I
·
X
=
X
·
I
=
0 and A is contractible modulo I, there exists ϕ
∈
X such that D
∘
π
=
a
d
ϕ on A
∖
I. Now if a
∈
I, clearly D
(
a
+
I
)
=
a
d
ϕ
(
a
) and if a
∈
A
∖
I then
(18)
D
(
a
+
I
)
=
D
∘
π
(
a
)
=
a
d
ϕ
(
a
)
=
a
·
ϕ
-
ϕ
·
a
.
Thus D
=
a
d
ϕ.
(iii) It follows from (ii).
(iv) This follows from (ii) and Exr 4.1.1 [9].
(v) It follows from (iv) and Lemma11.Theorem 13.
A is contractible modulo I if and only if A has a diagonal modulo I.Proof.
SinceA is contractible modulo I, A
/
I has an identity e
~ (by Theorem 12). Let E
≔
e
~
⊗
e
~
∈
(
A
/
I
)
⊗
^
(
A
/
I
) and a
d
E
:
A
→
(
A
/
I
)
⊗
^
(
A
/
I
) by a
↦
a
·
E
-
E
·
a; then
(19)
π
A
/
I
(
a
·
E
-
E
·
a
)
=
0
(
a
∈
A
)
.
So a
d
E
(
a
)
⊆
ker
⁡
π
A
/
I
(
a
∈
A
). Since I
·
ker
⁡
π
A
/
I
=
ker
⁡
π
A
/
I
·
I
=
0, contractibility of A
/
I implies that there exists N
∈
ker
⁡
π
A
/
I such that a
d
E
=
a
d
N on A
∖
I. Define M
≔
E
-
N; then
(20)
a
·
π
A
/
I
(
M
)
=
a
·
e
~
2
-
0
=
a
+
I
(
a
∈
A
)
,
a
·
M
-
M
·
a
=
a
·
(
E
-
N
)
-
(
E
-
N
)
·
a
=
a
d
E
(
a
)
-
a
d
N
(
a
)
=
0
(
a
∈
A
∖
I
)
.
Conversely, LetM be a diagonal modulo I, let E be a Banach A-bimodule such that I
·
E
=
E
·
I
=
0, and let D
:
A
→
E be a bounded derivation. Put M
≔
∑
n
=
1
∞
‍
a
n
′
⊗
b
n
′, where a
n
′
=
a
n
+
I and b
n
′
=
b
n
+
I and ∑
n
=
1
∞
‍
∥
a
n
∥
∥
b
n
∥
<
∞. Set x
≔
∑
n
=
1
∞
‍
a
n
·
D
(
b
n
); then x
∈
E. Let F
:
A
→
E by F
(
a
)
=
a
·
x
-
x
·
a. Then F is an inner derivation and F
(
a
)
=
0
(
a
∈
I
). Let a
∈
A
∖
I
,
ϕ
∈
E
*; then
(21)
〈
a
·
x
,
ϕ
〉
=
〈
∑
n
=
1
∞
‍
a
a
n
D
(
b
n
)
,
ϕ
〉
=
〈
∑
n
=
1
∞
‍
a
n
D
(
b
n
a
)
,
ϕ
〉
=
〈
∑
n
=
1
∞
‍
a
n
D
(
b
n
)
·
a
,
ϕ
〉
+
〈
∑
n
=
1
∞
‍
a
n
b
n
D
(
a
)
,
ϕ
〉
=
〈
x
·
a
,
ϕ
〉
+
〈
D
(
a
)
,
ϕ
〉
.
Thus a
·
x
=
x
·
a
+
D
(
a
), so D
=
F on A
∖
I.By replacingX instead of X
* in the Theorem 8, we have the following theorem.Theorem 14.
LetA be contractible modulo I, let B be a Banach algebra, and let J be a closed ideal of B. Let θ
:
A
→
B be a continuous homomorphism with dens range such that θ
(
I
)
⊆
J. Then B is contractible modulo J.


## 4. Contractibility Modulo an Ideal of Semigroup Algebra

In this section we state a version of Selivanov's theorem [9] for a large class of semigroups, characterizing contractibility modulo an ideal for the semigroups. Let S be a semigroup and let E
(
S
) be the set of idempotents of S. A congruence ρ on semigroup S is called a group congruence if S
/
ρ is a group. We denote the least group congruence on S by σ, as used in [10, 11]. We now recall the following lemma of [6].Lemma 15.
LetS be a semigroup:(i)
for a group congruenceρ on S, l
1
(
S
/
ρ
)
≃
l
1
(
S
)
/
I
ρ, where I
ρ is a closed ideal of l
1
(
S
),
(ii)
ifσ is the least group congruence on S and e
f
=
f
e
(
e
,
f
∈
E
(
S
)
), then I
σ
=
I
σ
2.A semigroupS is called E-inversive if for all x
∈
S there exists y
∈
S such that x
y
∈
E
(
S
). Let S be a semigroup, let Reg
(
S
)
=
{
a
∈
S
:
a
∈
a
S
a
} be the set of regular elements of S, and let V
(
a
)
=
{
x
∈
S
:
a
=
a
x
a
,
x
=
x
a
x
} be the set of inverses of a
∈
S. When V
(
a
)
≠
ϕ, for every a
∈
S, S is calledregular. Let W
(
a
)
=
{
x
∈
S
:
x
=
x
a
x
} be the set of all weak inverses of element a
∈
S. Then S is E-inversive if and only if W
(
a
)
≠
ϕ for all a
∈
S [12]. In particular, regular semigroups are E-inversive. We recall a semigroup S is called an E-semigroup if E
(
S
) forms a subsemigroup of S. We now recall the following preposition of [13].Proposition 16.
IfS is an E-inversive E-semigroup with commutative idempotents, then the relation σ
=
{
(
a
,
b
)
∈
S
×
S
∣
e
a
=
f
b
f
o
r
s
o
m
e
e
,
f
∈
E
S
} is the least group congruence on S.A semigroupS is called eventually regular if every element of S has some power that is regular and E
(
S
) is semilattice. It is shown that if S is an eventually semigroup then the relation σ
=
{
(
s
,
t
)
:
e
s
=
e
t
,
for
some
e
∈
E
(
S
)
} is the least group congruence on S [14].Theorem 17.
IfS is either(i)
E-inverse E-semigroup with commutative idempotents or
(ii)
eventually inverse semigroup with commutative idempotents,then l
1
(
S
) is contractible modulo I
σ if and only if S
/
σ is finite.Proof.
By Selivanov's theorem [7], S
/
σ is finite if and only if l
1
(
S
/
σ
)
≃
l
1
(
S
)
/
I
σ is contractible. Now by using Theorem 12 proof is complete.Example 18.
SupposeN is commutative semigroup of positive integers with maximum operation, then E
(
N
)
=
N. Suppose m
σ
n if and only if k
m
=
k
n, for some k
∈
E
(
N
)
(
n
,
m
∈
N
). Then σ is the least group congruence on N and N
/
σ
≃
G
N is the trivial maximum group image of N. Since G
N
≃
N
/
σ is finite, l
1
(
N
/
σ
) is contractible and by Lemma 15 and Theorem 12,  l
1
(
N
) is contractible modulo I
σ. We note that l
1
(
N
) is not contractible because l
1
(
N
) has no diagonal. In fact, if l
1
(
N
) has a diagonal, so it should be form m
=
δ
1
⊗
δ
1. Then δ
k
·
m
=
m
·
δ
k
(
k
∈
N
). But this equality holds if and only if δ
k
⊗
δ
1
=
δ
1
⊗
δ
k which is a contradiction.Example 19.
LetC
=
{
p
m
q
n
:
m
,
n
≥
0
} be the bicyclic semigroup generated by p
,
q. Clearly, C is an E-unitary semigroup with E
(
C
)
=
{
p
n
q
n
:
n
=
0,1
,
2
,
…
}. Then C
/
σ
=
G
C
=
Z, where σ is defined by x
σ
y if and only if e
x
=
e
y for some e
∈
E
(
C
). Since G
C
=
Z is infinite, l
1
(
G
C
)
≅
l
1
(
C
)
/
I
ρ is not contractible. Then l
1
(
C
) is not contractible modulo I
σ (by Theorem 12 (ii)). But l
1
(
C
) is amenable modulo I
σ [6].



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*Source: 514761-2014-07-07.xml*