#### 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, including *E*-inversive *E*-semigroup and eventually inverse semigroups.

#### 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 is amenable (in the usual sense) if and only if (resp., ) 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 modulo ideals by certain classes of group congruences on is equivalent to the amenability of , 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 algebra is contractible modulo an ideal if and only if is finite, restoring Selivanov's theorem for a large class of semigroups [7].

The paper is organized as follows. In Section 2 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

Let be a Banach algebra and let be a Banach -bimodule. By a derivation of to we mean a bounded linear map such that . Clearly the mapping defined by is a derivation that is called inner (). A Banach algebra is called amenable if, for any -bimodule and any derivation , there exists such that .

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. *Let be a closed ideal of . A Banach algebra is amenable modulo if for every Banach -bimodule such that and every derivation from into there is such that on the set theoretic difference .

Proposition 2 (see [6, Theorem 1]). *Let be a closed ideal of .*(i)*If is amenable and , then is amenable modulo .*(ii)*If is amenable modulo , then is amenable.*(iii)*If is amenable modulo and is amenable, then is amenable.*

*Blanket Assumption*. All over this paper we fix and as above, unless they are otherwise specified.

*Definition 3. *Let be a Banach algebra. A bounded net is approximate identity modulo if

Theorem 4. *Let be amenable modulo . Then has an approximate identity modulo .*

*Proof. *Set ; then is a Banach -bimodule where it acts on via
Clearly . Suppose , where is a natural inclusion map and is a natural quotient map. Since is a derivation, there exists such that . Goldstine's theorem provides a bounded net in such that . Then . Now
which is equivalent to . Passing to convex combinations, we can suppose that . In an analogous fashion, we obtain a net in such that
Define . Then is an approximate identity modulo .

We now express the version of Cohen's factorization theorem [8], where has an approximate identity modulo . Since the proof is similar, we omit the proof.

Theorem 5. *If has a bounded approximate identity modulo , , and , then there exist such that and .*

Let be a Banach algebra and let be a closed ideal of ; then the corresponding diagonal operator of is defined as Clearly, is a bimodule homomorphism.

*Definition 6. *(i) An element is a virtual diagonal modulo if

(ii) A bounded net is an approximate diagonal modulo if

(iii) An element is a diagonal modulo if

Theorem 7. *The following conditions are equivalent:*(i)* is amenable modulo ,*(ii)*there is an approximate diagonal modulo ,*(iii)*there is a virtual diagonal modulo .*

*Proof. *(i) ⇒ (iii) Since is amenable modulo , is amenable. Thus has a bounded approximate identity . Let be a -accumulation point of . Put ; then
Theorem 5 implies that the map is surjective; then . Furthermore can be made into Banach -bimodule by defining
Since is a closed submodule of , is a dual Banach -bimodule. Let and ; then . Since , amenability modulo of implies that there exists such that on . Define ; then

(ii) ⇒ (i) Let , where and with . Suppose is a Banach -bimodule such that and is a bounded derivation. Then is a bounded net in , which has a -accumulation point; say . Let by . Clearly, is an inner derivation and . Let and ; then
So and on .

(iii) ⇒ (ii) Let be a virtual diagonal modulo and let be a bounded net in with . Then
By passing to convex combinations, we obtain an approximate diagonal.

Theorem 8. *Let be amenable modulo , let be a Banach algebra, and let be a closed ideal of . Let be a continuous homomorphism with dens range such that . Then is amenable modulo .*

*Proof. *Suppose is a Banach -bimodule such that and is a bounded derivation. Then becomes a Banach -bimodule via
Clearly, and is a bounded derivation. Since is amenable modulo , there exists such that on . Let ; then there is a net such that . Since , we may assume that . We have
Thus is amenable modulo .

#### 3. Contractibility Modulo an Ideal of Banach Algebra

We recall that a Banach algebra is contractible (super-amenable) if for every Banach -bimodule , where the left hand side is the first cohomology group of with coefficient in (see [3, 6]).

*Definition 9. *A Banach algebra is contractible modulo if for every Banach -bimodule such that , every bounded derivation from into is an inner derivation on the set theoretic difference .

Clearly, if is contractible modulo , then is amenable modulo .

*Definition 10. *An element is an identity modulo if . A Banach algebra is unital modulo if has an identity modulo .

Lemma 11. * has an identity modulo if and only if has an identity.*

*Proof. *The element is an identity modulo of if and only if . These relations are equivalent to or is an identity of .

Theorem 12. *The following assertions hold.*(i)*If is contractible and , then is contractible modulo .*(ii)*If is contractible modulo , then is contractible.*(iii)*If A is contractible modulo and is contractible, then is contractible.*(iv)*If is contractible modulo , then is unital.*(v)*If is contractible modulo , then is unital modulo .*

*Proof. *(i) Let be a Banach -bimodule such that and let be a bounded derivation. Since , with module action can be regarded as -bimodule. Let by . Since , . Hence is well-defined bounded derivation. Since is contractible, there exits such that . We have

(ii) Let be a Banach -bimodule and let be a bounded derivation. Now can be made into Banach -bimodule by defining
where is the natural quotient map. Define . Clearly, is bounded derivation. Since and is contractible modulo , there exists such that on . Now if , clearly and if then
Thus .

(iii) It follows from (ii).

(iv) This follows from (ii) and Exr 4.1.1 [9].

(v) It follows from (iv) and Lemma 11.

Theorem 13. * is contractible modulo if and only if has a diagonal modulo .*

*Proof. *Since is contractible modulo , has an identity (by Theorem 12). Let and by ; then
So . Since , contractibility of implies that there exists such that on . Define ; then

Conversely, Let be a diagonal modulo , let be a Banach -bimodule such that , and let be a bounded derivation. Put , where and and . Set ; then . Let by . Then is an inner derivation and . Let ; then
Thus , so on .

By replacing instead of in the Theorem 8, we have the following theorem.

Theorem 14. *Let be contractible modulo , let be a Banach algebra, and let be a closed ideal of . Let be a continuous homomorphism with dens range such that . Then is contractible modulo .*

#### 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 be a semigroup and let be the set of idempotents of . A congruence on semigroup is called a group congruence if is a group. We denote the least group congruence on by , as used in [10, 11]. We now recall the following lemma of [6].

Lemma 15. *Let be a semigroup:*(i)*for a group congruence on , , where is a closed ideal of ,*(ii)*if is the least group congruence on and , then .*

A semigroup is called -*inversive* if for all there exists such that . Let be a semigroup, let be the set of regular elements of , and let be the set of inverses of . When , for every , is called* regular*. Let be the set of all weak inverses of element . Then is -inversive if and only if for all [12]. In particular, regular semigroups are -inversive. We recall a semigroup is called an -*semigroup* if forms a subsemigroup of . We now recall the following preposition of [13].

Proposition 16. *If is an -inversive -semigroup with commutative idempotents, then the relation is the least group congruence on S.*

A semigroup is called eventually regular if every element of has some power that is regular and is semilattice. It is shown that if is an eventually semigroup then the relation is the least group congruence on [14].

Theorem 17. *If is either*(i)*-inverse -semigroup with commutative idempotents or*(ii)*eventually inverse semigroup with commutative idempotents,**then is contractible modulo if and only if is finite.*

*Proof. *By Selivanov's theorem [7], is finite if and only if is contractible. Now by using Theorem 12 proof is complete.

*Example 18. *Suppose is commutative semigroup of positive integers with maximum operation, then . Suppose if and only if , for some . Then is the least group congruence on and is the trivial maximum group image of . Since is finite, is contractible and by Lemma 15 and Theorem 12, is contractible modulo . We note that is not contractible because has no diagonal. In fact, if has a diagonal, so it should be form . Then . But this equality holds if and only if which is a contradiction.

*Example 19. *Let be the bicyclic semigroup generated by . Clearly, is an -unitary semigroup with . Then , where is defined by if and only if for some . Since is infinite, is not contractible. Then is not contractible modulo (by Theorem 12 (ii)). But is amenable modulo [6].

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.