International Journal of Analysis and Applications
ISSN 2291-8639
Volume 13, Number 2 (2017), 178-184
http://www.etamaths.com

SOME GENERALIZED NOTIONS OF AMENABILITY MODULO AN IDEAL

HOSEIN ESMAILI AND HAMIDREZA RAHIMI∗

Abstract. In this paper some generalized notions of amenability modulo an ideal of Banach al-
gebras such as uniformly (boundedly) approximately amenable (contractible) modulo an ideal of

Banach algebras are investigated. Using the obtained results, uniformly (boundedly) approximately

amenability (contractibility) modulo an ideal of weighted semigroup algebras are characterized.

1. Introduction

Let A be a Banach algebra and X be a Banach A-bimodule, by a derivation D we mean a bounded
linear map D : A → X such that D(ab) = a.D(b) + D(a).a, (a,b ∈ A). An inner derivation is
a derivation D which there exists x ∈ X such that D(a) = adx(a) = a · x − x · a, (a ∈ A). A
Derivation D : A → X is called approximately inner if there exists a net (ξα) in X such that D(a) =
lim
α
adξα(a) (a ∈ A) where the limit is taken in norm of X. If the above limit exists in the w∗-

topology (say, X is a dual module) then D is called w∗-approximately inner. A Banach algebra A is
called boundedly approximately amenable (contractible) if, for each Banach A-bimodule X and each
continuous derivation D : A → X∗ (D : A → X) there exist K > 0 and a net (ξα) in X∗ (in X) such
that for each a ∈ A and α, ‖a.ξα − ξα.a‖ ≤ M.‖a‖ and D(a) = limα adξα(a), A is called uniformly
approximately amenable (contractible) if for each Banach A-bimodule X, each continuous derivation
D from A to X∗ ( to X) is the limit of a sequence of inner derivations in the norm topology of the set
of all bounded operators from A into X∗, i.e. B(A,X∗) (into X, i.e. B(A,X)). Some characterizations
of these concepts of amenability are investigated in [5–7].

The concept of amenability modulo an ideal for a class of Banach algebras which could be considered
as a generalization of amenability of Banach algebra was introduced by the first author and Amini in
2014 [1]. Using this idea, it is shown that a semigroup S is amenable if and only if the semigroup
algebra l1(S) is amenable modulo an ideal induced by appropriate congruence σ on S, for a large
class of semigroups. In further researches, it was shown that amenability modulo an ideal can be
characterized by the existence of virtual diagonal modulo an ideal and approximate diagonal modulo
an ideal. To see the details of these results and more on this topic, we refer to [1, 10, 11].

In this paper we shall continue the investigation of amenability modulo an ideal, in particular
that of boundedly approximate amenability modulo an ideal and uniformly approximate amenability
modulo an ideal of Banach algebras. Afterward, for a large class of semigroups, we introduce some
characterization of amenability modulo an ideal of weighted semigroup algebras.

This paper is organized as follow; in section two, we give some basic notions of generalized amenabil-
ity and amenability modulo an ideal of Banach algebras and we show that the concepts approxi-
mately contractible modulo an ideal, approximately amenable modulo an ideal and w∗−approximately
amenable modulo an ideal of Banach algebras are equivalent. In section three, we investigate to the gen-
eralized notions of amenability modulo an ideal of Banach algebras such as, uniformly approximately
amenable (contractible) modulo an ideal and boundedly approximately amenable (contractible) mod-
ulo an ideal of Banach algebras. In section four, we consider the generalized notions of amenability

Received 19th November, 2016; accepted 16th January, 2017; published 1st March, 2017.
2010 Mathematics Subject Classification. 43A07, 46H25.
Key words and phrases. uniformly approximately amenable modulo an ideal; boundedly approximately amenable

modulo an ideal; weight semigroup algebra.

c©2017 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

178



SOME GENERALIZED NOTIONS OF AMENABILITY MODULO AN IDEAL 179

modulo an ideal for the weighted semigroup algebra l1(S) and we finish this section with give some
examples.

2. preliminaries

In this section we recall some basic notions which we need in this paper. To see more details, reader
can refer to [1, 10–12].

Definition 2.1. Let I be a closed ideal of A. A Banach algebra A is amenable (contractible) modulo
I if for every Banach A-bimodule X such that I ·X = X ·I = 0, and every derivation D from A into
X∗ (into X) there is φ ∈ X∗ such that D = adφ on the set theoretic difference A\I := {a ∈ A : a /∈ I}.

All over this paper we fix A and I as above, unless they are otherwise specified.

Theorem 2.1. ( [1, Theorem 1]) The following assertions hold.
i) If A/I is amenable and I2 = I then A is amenable modulo I.
ii) If A is amenable modulo I then A/I is amenable.
iii) If A is amenable modulo I and I is amenable, then A is amenable.

Let A be a Banach algebra and I be a closed ideal of A. With the module actions a.b̄ :=
ab and b̄.a := ba, A

I
is a Banach A-bimodule where ā is the image of a in A

I
. Also A

I
⊗̂A can be

consider as a Banach A−bimodule where the module actions are the linear extension of a.(b̄ ⊗ c) :=
ab⊗c and (b̄⊗c).a := (b̄⊗ca), (a,b,c ∈ A). By the diagonal operator we mean the bounded linear op-
erator defined by the linear extension of π : (A

I
⊗̂A) → A

I
by π(b̄⊗c) = bc. Clearly, π is a A−bimodule

homomorphism.

Definition 2.2. (i) By a virtual diagonal modulo I, we mean an element M ∈ (A
I
⊗̂A)∗∗ such that;

a ·π∗∗(M) − ā = 0 (a ∈ A) and a ·M −M ·a = 0 (a ∈ A\ I),
(ii) an approximate diagonal modulo I, we mean a bounded net (mα)α ⊆ (AI ⊗̂A) such that;

a.π(mα) − ā → 0 (a ∈ A) and a.mα −mα.a → 0 (a ∈ A\ I).
(iii) a diagonal modulo I, we mean an element m ∈ (A

I
⊗̂A) such that;

a.π(m) − ā = 0 (a ∈ A), and a.m−m.a = 0, (a ∈ A\ I).

We recall that a bounded net (uα)α ⊆ A is called approximate identity modulo I if lim
α
uα · a =

lim
α
a ·uα = a (a ∈ A\I). If A is amenable modulo I then A has an approximate identity modulo I. It

is shown that a Banach algebra A is amenable modulo I if and only if A has an approximate diagonal
modulo I,if and only if A has a virtual diagonal modulo I [10]. By appropriate modifications, the
following Theorem may be proved in much the same way as [4, Theorem 1.9.21].

Theorem 2.2. A is contractible modulo I if and only if A has a diagonal modulo I.

Definition 2.3. A Banach algebra A is called approximately amenable (contractible) modulo I if for
every Banach A-bimodule X such that I · X = X · I = 0, every bounded derivation D : A → X∗
(D : A → X) is approximately inner on the set theoretical difference A\I := {a ∈ A : a /∈ I}.

Theorem 2.3. The following statements are equivalent;
a) A is approximately contractible modulo I;
b) A is approximately amenable modulo I;
c) A is w∗-approximately amenable modulo I.

Proof. It is easily seen that (a → b) and (b → c), so we only need to show that (c → a). Since
A is w∗-approximately amenable modulo I, A] is w∗-approximately amenable modulo I ( by [11,

Theorem 3.2]). Now [11, Theorem 3.3], provide us to consider a net (Mi) ⊆ (A
]

I
⊗̂A])∗∗ such that

a · Mi − Mi · a → 0 (∀a ∈ A]\I) and π∗∗(Mi) → ē in the w∗-topology of (A
]

I
⊗̂A])∗∗ and A∗∗,

respectively. Let � > 0 and consider finite sets F ⊆ A]\I, Φ ⊆ (A]\I)∗ and N ⊆ (A
]

I
⊗̂A])∗, so there

exists j such that (a ∈F,φ ∈ Φ,f ∈N),
|〈a.f −f.a,Mj〉| = |〈f,a.Mj −Mj.a〉| < � and | 〈φ,π∗∗(Mj) − ē〉 |< �



180 ESMAILI AND RAHIMI

Using the weak∗-continuity of π∗∗ and Goldstine’s theorem, we can choose m ∈ (A
]

I
⊗̂A]) such that

| 〈f,a.m−m.a〉 |=| 〈a.f −f.a,m〉 |< �, and | 〈φ,π(m) − ē〉 |< �,

for each a ∈F,φ ∈ Φ and f ∈N . Hence there exists (mi) ⊆ (A
]

I
⊗̂A]) such that a.mi−mi.a → 0 (a ∈

A\I) and π(mi) → ē in the w-topology of (A
]

I
⊗̂A]) and A], respectively. Now for every finite set

F = {a1,a2, , ...,an}⊆ A]\I,

(a1.mi −mi.a1, ...,an.mi −mi.an,π(mi)) → (0, ..., 0, ē)

weakly in (A
]

I
⊗̂A]) ⊕ (A]\I). Therefore

(0, ..., 0, ē) ∈ c̄ow{(a1.mi −mi.a1, ...,an.mi −mi.an,π(mi))}.

Set P = {(a1.mi −mi.a1, ...,an.mi −mi.an,π(mi))}, so

co(P) = {(a1.M −M.a1, ...,an.M −M.an,π(M)) ∈ co{mi}}.

We have

(0, ..., 0, ē) ∈ c̄ow(P) = c̄o‖‖(P).
The Hahn-Banach theorem implies that for each � > 0 there exists u�,F ∈ co{mi} such that

‖a.u�,F −u�,F .a‖ < � and ‖π(u�,F ) − ē‖ < �, (a ∈ F).

Now by [11, Theorem 3.8] proof is complete. �

3. Uniformly and boundedly approximate amenability (contractibility) modulo an
ideal of Banach algebras

Definition 3.1. A Banach algebra A is uniformly approximately amenable (contractible) modulo I
if for every Banach A-bimodule X such that I · X = X · I = 0 and every continuous derivation
D : A → X∗ (D : A → X) there is a net (xα) ⊆ X∗ ((xα) ⊆ X) such that D(a) = lim

α
adxα(a) where

the convergence is uniform for each a ∈ A\I such that ‖a‖≤ 1,

Lemma 3.1. A Banach algebra A is uniformly approximately contractible modulo I if and only if A]

is uniformly approximately contractible modulo I.

Proof. Let A be uniformly approximately contractible modulo I, X be a Banach A]−bimodule and
D : A] → X be a bounded derivation. Then there are ξ ∈ eXe and D1 : A] → eXe such that D =
D1 +adξ. We have D1(e) = 0 and D1|A ∈Z1(A,eXe). Since A is uniformly approximately contractible
modulo I, there exits (ζn) ∈ eXe such that D1(a,α) = lim

n
adζn(a), (a ∈ A\I,α ∈ C,‖a‖ + |α| ≤ 1).

Now if (a,α) ∈ (A\I) ⊕C = (A\I)] such that ‖a‖ + |α| ≤ 1, then D1(a,α) = D1(a, 0) + αD1(e) =
D1(a) = adζa(a). Hence D(a) = D1(a) + adξ(a) = lim

n
adζn(a) + adξ(a) = lim

n
adζn+ξ(a).

Conversely, let X be a Banach A−bimodule and D : A → X∗ be a bounded derivation. Defining
(a,α).x = a.x + α.x and x.(a,α) = x.a + α.x (x ∈ X, (a,α) ∈ A]) makes A] into an A]−bimodule.
Define D̃ : A] → X by D̃(a,α) = D(a) ((a,α) ∈ A]). Clearly D̃ is a bounded derivation. Supposing A]
is uniformly approximately contractible modulo I, there is (ξn) ⊆ X such that D̃ = lim

n
adξn on the

unit ball of (A\I)]. Now D̃|A, as required. �

Lemma 3.2. Let X be an A-module and (en) ⊆ X be a sequence such that for each a ∈ A\I with
‖a‖ ≤ 1, a = lim

n
a.en. Then A has a right identity modulo I, i.e. there exists u ∈ A such that

a.u = a (a ∈ A\I).

Proof. Let Rf denote the right multiplication by f ∈ X. Then there is (en) ⊆ X with ‖Rf − id‖≤ 1,
so Rf is invertible. This implies that there is a g ∈ B(X,A) such that Rf ◦ g = id. Set u = g(f), so
u.f = Rf ◦ g(f) = f. Then auf = af and for each a ∈ A\I, (au−a).f = 0. This means that u is a
right identity modulo I. �

Lemma 3.3. Suppose that A is uniformly approximately contractible modulo I. Then A has an identity
e on A\I, i.e. e.a = a.e = a (a ∈ A\I).



SOME GENERALIZED NOTIONS OF AMENABILITY MODULO AN IDEAL 181

Proof. Consider A as a A−bimodule where the module actions are defined by a.x = ax and x.a =
0 (a ∈ A,x ∈ X). Let D : A → A∗∗ defined by D(a) = â be the canonical embedding. It is clear
that D is a bounded derivation. Since A is uniformly approximately contractible modulo I, there is
(eα) ⊆ A∗∗ such that D(a) = lim aden(a) (a ∈ A\I,‖a‖ ≤ 1), so a = lim

n
a.en. Using Lemma 3.2, A

has a right identity modulo I. The same argument is true for Aop, and hence A has an identity e on
A\I. �

Theorem 3.1. Let A be uniformly approximately contractible modulo I. Then A is contractible modulo
I.

Proof. By Lemma 3.3, we may suppose that A has an identity ”e” on A\I. Define D : A → kerπ ⊆
(A
I
⊗̂A) by D(a) = ā⊗e− ē⊗a. Then D is a bounded derivation and ‖D‖≤ 2. Since A is uniformly

approximately contractible modulo I, there is (tn) ∈ kerπ such that adtn → D uniformly for a ∈ A\I,
with ‖a‖ ≤ 1. Suppose that tn =

∑
i

x̄ni ⊗ y
n
i and s =

∑
i

āj ⊗ bj ∈ kerπ. Since π(s) = π(tn) = 0,∑
i

āibi =
∑
i

aibi = 0 and
∑
i

x̄nj y
n
j =

∑
i

xnj y
n
j = 0. Hence,

‖stn −s‖ = ‖
∑
i,j

ājx̄
n
i ⊗y

n
i bj −

∑
j

āj ⊗ bj‖

= ‖
∑
i,j

ājx̄
n
i ⊗y

n
i bj −

∑
i,j

ājbjx̄
n
i ⊗y

n
i −

∑
j

āj ⊗ bj +
∑
j

ājbj ⊗e‖

= ‖
∑
j

āj(
∑
i

x̄ni ⊗y
n
i bj −

∑
i

bjx̄
n
i ⊗y

n
i − ē⊗ bj + bj ⊗e)‖

≤
∑
j

‖
∑
i

x̄ni ⊗y
n
i bj −

∑
i

bjx̄
n
i ⊗y

n
i −e⊗ bj + bj ⊗e‖‖āj‖

=
∑
j

‖
∑
i

x̄ni ⊗y
n
i

bj
‖bj‖

−
∑
i

bj
‖bj‖

x̄ni ⊗y
n
i

−e⊗ bj‖bj‖ +
bj
‖bj‖
⊗e‖‖āj‖‖bj‖

≤
∑
j

sup
‖c‖≤1

‖tn.c− c.tn −e⊗ c + c⊗e‖‖āj‖‖bj‖.

It implies that ‖stn − s‖ ≤ sup
‖c‖≤1

‖adtn(c) −D(c)‖ on the unit ball of kerπ, hence stn → s uniformly

on the unit ball of kerπ and by Lemma 3.2, kerπ has a right identity modulo I, u. Set v = ē⊗e−u,
then π(v) = ē−π(u) and for each a ∈ A\I, a.v−v.a = 0. Thus v is a diagonal modulo I and hence A
is contractible modulo I( by Theorem 2.2). �

Definition 3.2. A Banach algebra A is boundedly approximate amenable (contractible) modulo I if
for each Banach A-bimodule X with X · I = I · X = 0 and each continuous derivation D : A → X∗
(D : A → X) there exist K > 0 and a net (ξα) in X∗ (X) such that for each a ∈ A\I and α,
‖a.ξα − ξα.a‖≤ M.‖a‖, and D(a) = limα adξα(a).

Theorem 3.2. Then the following assertions hold;
(i) if A is boundedly approximate amenable modulo I, then A

I
is boundedly approximate amenable.

(ii) if A
I

is boundedly approximate amenable and I2 = I then A is boundedly approximate amenable
modulo I

Analogous assertions satisfy for uniformly approximately amenable modulo an ideal Banach algebras.

Proof. (i) Suppose that X is a Banach A
I

-bimodule and D : A
I
→ X∗ is a bounded derivation.

Now X is a clearly Banach A- module with the module actions defined by a.x = π(a).x , x.a =
x.π(a), (a ∈ A,x ∈ X) where π : A → A

I
is the canonical quotient map. Since I ·X = X · I = 0 and

D ◦π : A → X∗ is a bounded derivation, there is a (ξα) ⊂ X∗ such that ‖a.ξα − ξα.a‖ ≤ M.‖a‖ (for
some M > 0) and D◦π = lim

α
adξα on A\I. We have ‖π(a).ξα−ξα.π(a)‖ = ‖a.ξα−ξα.a‖≤ M.‖a‖ and

D(π(a)) = D ◦π(a) = lim
α
adξα(a), (π(a) ∈

A

I
). Hence A

I
is boundedly approximate amenable modulo

I.



182 ESMAILI AND RAHIMI

(ii) Suppose that X is a Banach A-bimodule such that X · I = I · X = 0 and D : A → X∗ is a
bounded derivation. We can consider X as an A

I
-bimodule with the module actions a.x = π(a).x , x.a =

x.π(a), (a ∈ A,x ∈ X). The equality I2 = I provide us to define the well-defined bounded derivation
D : A

I
→ X∗ by D(π(a)) = D(a) (a ∈ A). Since A

I
is boundedly approximate amenable modulo I,

there is a (ξα) ⊂ X∗ such that ‖π(a).ξα−ξα.π(a)‖≤ M.‖a‖ (for some M > 0) and D = limα adξα. It is
not far to see that the net (adξα) is norm bounded in B(A,X∗) and D(a) = D(π(a)) = lim

α
adξα(a). �

The proof of the following result is the same way as Theorem 3.2.

Corollary 3.1. The following conditions are hold;
(i) if A is boundedly approximate contractible modulo I, then A

I
is boundedly approximate con-

tractible.
(ii) if A

I
is boundedly approximate contractible and I2 = I then A is boundedly approximate con-

tractible modulo I
Analogous assertions satisfy for uniformly approximately contractible modulo an ideal.

For a Banach algebra A, it is shown that A is uniformly approximately amenable if and only if it is
amenable [6, Theorem 3.1]. Using Theorem 3.2, we have the following result.

Corollary 3.2. Suppose A is a Banach algebra and I is a closed ideal of A such that I2 = I. Then A
is uniformly approximate amenable modulo I if and only if it is amenable modulo I.

Theorem 3.3. A Banach algebra A is boundedly approximate amenable modulo I if and only if there
exists a constant M > 0 such that for any Banach A-bimodule X with X · I = I · X = 0 and any
continuous derivation D : A → X∗ there is a net (ηi) ⊆ X∗ such that
a) sup

i
‖adηi‖≤ M‖D‖,

b) D(a) = lim
i
adηi(a), (∀a ∈ A\I).

Proof. Let assumptions (a) and (b) hold, then ‖adηi‖ ≤ M‖D‖ =
M‖D‖
‖a‖ (a ∈ A/I). Therefore A

is boundedly approximately amenable modulo I. Conversely, let A be a boundedly approximately
amenable modulo I. Consider there is no such M. Suppose that for every integer n ∈ N, Mn is
Banach module such that Mn · I = I · Mn = 0 and Dn : A → M∗n is a derivation with ‖Dn‖ > n.
Now X = l1(Mn) is a Banach A−module with dual l∞(M∗n). Put D = (Dn), D : A → l∞(M∗n) is
a continuous derivation and D(a) = (Dn(a)) = lim

i
(adηn

i
(a)). Since ‖Dn‖ > n, ‖D‖ → ∞ which is

contradiction. �

The same argument of [12, Theorem 3.2 and 3.3] and minor changes, we have the following theorems;

Theorem 3.4. A Banach algebra A is boundedly approximately amenable modulo I if and only if A#

is boundedly approximately amenable modulo I.

Theorem 3.5. Let A be a Banach algebra and I be a closed ideal of A. If A is boundedly approximately
amenable modulo I then;

(a) there is a net (Mi)i ⊆ (A
]

I
⊗̂A#)∗∗ and a constant L > 0 such that ā.Mi−Mi. ā → 0, π∗∗(Mi) → ē,

and ‖ā.Mi −Mi. ā‖≤ L‖ā‖ , for each ā ∈ (A
#

I
).

Conversely, if (a) holds and the net (π∗∗(Mi)) is bounded then A is boundedly approximately amenable
modulo I.

4. Algebras related to discrete semigroups

We generally follow [3,9] for definitions and basic concepts of semigroups. For a semigroup S, the set
(possibly empty) of idempotents of S is denoted by E = E(S). A semigroup S is called an E-semigroup
if E(S) is a sub-semigroup of S, E-inversive if for each x ∈ S, there exists y ∈ S such that xy ∈ E(S),
regular if the set of inverses of a ∈ S, V (a) = {x ∈ S : a = axa,x = xax} 6= φ, inverse semigroup if
moreover, the inverse of each element is unique, E-unitary if for each x ∈ S and e ∈ E(S), ex ∈ E(S)
implies x ∈ E(S), semilattice if S is a commutative and idempotent semigroup and finally S is called
eventually inverse if every element of S has some power that is regular and E(S) is a semilattice.



SOME GENERALIZED NOTIONS OF AMENABILITY MODULO AN IDEAL 183

By a group congruence ρ on semigroup S we mean a congruence ρ such that S/ρ is a group. The
kernel of a congruence ρ on a semigroup S ”Kerρ” is the set {a ∈ S : aρ ∈ E(S/ρ)} = {a ∈ S :
(a,a2) ∈ ρ}. We denote the least group congruence on S (if exist) by σ. The least group congruence
on semigroups have also been considered by various authors [8, 13]. It is shown that if S is an E-
inversive E-semigroup such that E(S) is commutative (S is an eventually semigroup) then the relation
σ = {(a,b) ∈ S × S |ea = fb for some e,f ∈ ES} (σ′ = {(s,t) : es = et, for some e ∈ E(S)})
is the least group congruence on S [8, 13]. We recall that a function ω : G → (0,∞) such that
ω(g1g2) ≤ ω(g1)ω(g2) (g1,g2 ∈ G) is called a weight on group G. The weight ω on group G is called
symmetric if ω(g) = ω(g−1)(g ∈ G) and for any weight ω, by symmetrization of ω, we mean the weight
defined by Ωω(g) = ω(g)ω(g

−1). The weighted semigroup algebra (or Beurling algebra on semigroup
S) l1(S,ω) = {f | f : S → C,

∑
s∈S |f(s)|ω(s) < ∞} with ‖f‖1,ω =

∑
s∈S |f(s)|ω(s) and convolution

product is a Banach algebra. In the case ω = 1, the weighted semigroup algebra l1(S,ω) is called
semigroup algebra and is denoted by l1(S). We recall the following Lemma, which is detailed in [12].

Lemma 4.1. The following statements hold:

(i) if S is a semigroup, ρ is a congruence on S and ω is a weight on S, then
l1(S,ω)
Iρ

' l1(S/ρ,ωρ)
where ωρ([s]ρ) = inf{ω(s) : s ∈ [s]ρ} is the induced weight on S/ρ and Iρ is an ideal in l1(S,ω)
generated by the set

{δs − δt : s,t ∈ S with (s,t) ∈ ρ};
(ii) if S is an E-inversive semigroup with commuting idempotents or S is an eventually inverse

semigroup, σ is the least group congruence on S and ω is a weight on S, then l1(S/σ,ωσ)) '
l1(S,ω)
Iσ

where Iσ is a closed ideal of l
1(S,ω) and I2σ = Iσ.

It is shown that for a locally compact group G and a weight ω on G, the Beurling algebra L1(G,ω)
is boundedly approximately contractible if and only if the Beurling algebra L1(G,ω) is amenable, if
and only if G is amenable and Ω is bounded on G [7, Corollary 2.2]. The same conclusion can be
drawn for Beurling algebra of a weighted semigroup as follow;

Theorem 4.1. Suppose that ω is a weight on semigroup S. If S is an E-inversive semigroup with
commuting idempotents or S is an eventually inverse semigroup, then the followings assertions are
equivalent.

(i) The semigroup S is amenable and Ωωσ is bounded where ωσ is the induced weight on S/σ.
(ii) The weighted semigroup algebra l1(S,ω) is boundedly approximately contractible modulo Iσ.

Proof. The semigroup S is amenable if and only if S/σ is amenable [1, Theorem 2], if and only if
l1(S/σ,ωσ) is amenable (because S/σ is a group), if and only if l

1(S/σ,ωσ) is boundedly approximately
contractible (because Ωωσ is bounded on S/σ and by [7, Corollary 2.2]), if and only if l

1(S,ω) is
boundedly approximately contractible modulo Iσ (by Corollary 3.1). �

For a loccaly compact group G and a symmetric weight on ω on G, if lim
x→∞

ω(x) = ∞, then L1(G,ω)
is not boundedly approximately amenable [7, Corollary 2.8]. Thus we have the following corollary for
the weighted semigroup algebras;

Corollary 4.1. If S is a semigroup, ρ is a group congruence on S with Kerρ is central and ω
is a weight on semigroup S such that lim

x→∞
ω(x) = ∞(x ∈ S/ρ). Then l1(S,ω) is not boundedly

approximately amenable modulo Iρ.

Proof. Since Kerρ is central, the semigroup S is amenable if and only if S/ρ is amenable. On the other
hand, S/ρ is a group and lim

x→∞
ω(x) = ∞(x ∈ S/ρ), so l1(S/ρ,ωρ) is not boundedly approximately

amenable and consequensly l1(S,ω) is not boundedly approximately amenable modulo Iρ. �

We end this paper to give some illustrative examples.

Example 4.1. (i) Let S = {pmqn : m,n ≥ 0} be the bicyclic semigroup generated by p,q, then
S/σ ' Z where σ = {(s,t) ∈ S×S : se = te, for some e ∈ E(S)} is the least group congruence on S [1].
Using Theorem 4.1, amenability of S implies that l1(S) is boundedly approximately amenable modulo



184 ESMAILI AND RAHIMI

Iσ. We note that l
1(S) is not boundedly approximately amenable because l1(S) is a not approximate

amenable.
(ii) Let S = (N,∨) be the commutative semigroup of positive integers with maximum operation,

then E(S) = S. Set mσn if and only if km = kn, for some k ∈ E(S) (n,m ∈ N). Obviously σ
is the least group congruence on S and S/σ ' GS is the maximum group image of S. Since GS is
finite, l1(S/σ) is contractible and consequently l1(S/σ) is boundedly approximately contractible and
boundedly approximately amenable [2, 6]. Thus l1(S) is boundedly approximately contractible modulo
Iσ and boundedly approximately amenable modulo Iσ. We note that l

1(S) is not contractible because
l1(N) has not diagonal.

(iii) Let G = F2 be a free group with two generators a,b, T = (N0, +)×(N,max), where N0 = N∪{0}
and S = G×T . Then E(S) = {(1G,e) : e ∈ E(T)} is infinite. Under the homomorphism φ : (g,t) 7→ g,
G is the maximum group homomorphism image of S. Suppose that S/σ ' G where σ is a group
congruence on S. Then l1(S) is not boundedly approximately amenable (contractible) modulo Iσ,

since otherwise
l1(S)
Iσ
' l1(G) should be boundedly approximately amenable (contractible) which is

contradiction.

Acknowledgement. The authors sincerely thank the referee(s) for their valuable comments and
suggestions, which were very useful to improve the paper significantly.

References

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[11] H. Rahimi and E. Tahmasebi, Hereditary properties of amenability modulo an ideal of Banach algebras, J. Linear

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Department of Mathematics, Faculty of Science, Central Tehran Branch, Islamic Azad University, P.

O. Box 13185/768, Tehran, Iran

∗Corresponding author: rahimi@iauctb.ac.ir


	1. Introduction
	2. preliminaries
	3. Uniformly and boundedly approximate amenability (contractibility) modulo an ideal of Banach algebras
	4. Algebras related to discrete semigroups
	References