#### Abstract

In this paper, we investigate the existence of an ultra central approximate identity for a Banach algebra and its second dual . Also, we prove that for a left cancellative regular semigroup , has an ultra central approximate identity if and only if is a group. As an application, we show that for a left cancellative semigroup , is pseudo-contractible if and only if is a finite group. We also study this property for -Lau product Banach algebras and the module extension Banach algebras.

#### 1. Introduction and Preliminaries

A Banach algebra is amenable if there exists a bounded net in such that and for each , where is the projective tensor product with and is denoted for a bounded product morphism given by . It is well-known that every amenable Banach algebra has a bounded approximate identity . The existence of a bounded approximate identity may change the structure of a Banach algebra. For instance, the Fourier algebra with respect to a locally compact group has a bounded approximate identity if and only if is amenable (Leptin’s Theorem), see [1].

Some generalized notions of amenability like pseudo-contractibility and pseudo-amenability are given by Ghahramani and Zhang. A Banach algebra is called pseudo-contractible (pseudo-amenable) if there exists a not necessarily bounded net in such that and for each , respectively (see [2]). In fact, is a central approximate identity (approximate identity) with respect to the notion pseudo-contractibility (pseudo-amenability), respectively. We have to remind that for a locally compact group , the Segal algebra has a central approximate identity if and only if is a SIN group (see [3]).

Recently, the first author Pourabbas gave a notion of Johnson pseudo-contractibility for Banach algebras. In fact, a Banach algebra is Johnson pseudo-contractible if and only if there exists a not necessarily bounded net in such that and for each (see [4]). It is easy to see that is a net in such thatMotivated by these considerations in [5], the notion of ultra central approximate identity for Banach algebras is defined. In fact, a Banach algebra has an ultra central approximate identity if there exists a net in such that and , for every . In [5], the existence of an ultra central approximate identity for the semigroup algebra , where is a uniformly locally finite inverse semigroup, was characterized. They also showed that for the Brandt semigroup over a nonempty set , has an ultra central approximate identity if and only if is finite.

In this note, we study the existence of ultra central approximate identity for the second duals of certain semigroup algebras related to left cancellative semigroup. As an application, we show that for a left cancellative regular semigroup , is pseudo-contractible if and only if is a finite group. Some Banach algebras which do not have an ultra central approximate identity are given. We also investigate the property of ultra central approximate identity for -Lau product Banach algebras and the module extension Banach algebras.

#### 2. General Properties

Let be a Banach algebra. Throughout this note, the symbol is denoted for the second duals of a Banach algebra which is a Banach algebra with respect to the first Arens product. Jabbari et al. have introduced the notion of -inner amenability [6]. For a given Banach algebra , the set of all nonzero multiplicative linear functionals on is denoted by . Let , where . A Banach algebra is called -inner amenable if there exists a bounded linear functional on satisfying and for every and for every . This notion is equivalent with the existence of a bounded net in such that and , for every [7], [Proposition 2.2].

Theorem 1. *Let**be a Banach algebra and**. If**has an ultra central approximate identity, then**is**-inner amenable.*

*Proof. *Suppose that has an ultra central approximate identity. Then, there exists a net in such thatWe denote for unique extension of to is defined by for every and also is denoted for the unique extension of to is defined in the same way. Clearly and . So for every . Thus for a sufficient large , stays away from zero. Replacing with , we may assume that . By Goldstein’s theorem for every , there exists a net in such that in and . So by equation (2), in for every . Thus in for every . Let be an arbitrary finite subset of . Setwhere for every , . It is easy to see that is a convex set and is a -limit point of . Since , there exists a bounded net in such thatConsider the following order:So, the bounded net satisfiesBy Banach–Alaoglu’s theorem, has a -limit point say . One can see thatIt follows that is -inner amenable.

Using the similar argument to those in the proof of Theorem 1, we obtain the following corollary:

Corollary 1. *Let**be a Banach algebra and**. If**has an ultra central approximate identity, then**is**-inner amenable.*

*Example 1. *Let . With the finite -norm and matrix multiplication, becomes a Banach algebra. We claim that does not have an ultra central approximate identity. Arguing by contradiction, we suppose that has an ultra central approximate identity. Define by for every . It is easy to see that . Thus, by Theorem 1, is -inner amenable. By [6] and Theorem 1, there exists a bounded net in such thatfor all . Put instead of in (8), we have which is a contradiction. So, does not have an ultra central approximate identity.

*Example 2. *Let be a left zero semigroup with , that is, a semigroup with , for all . We claim that the semigroup algebra does not have an ultra central approximate identity. Suppose conversely that has an ultra central approximate identity. It is easy to see that , where is denoted for the augmentation character on . Applying Corollary 1, follows that is -inner amenable. Then, by [6] and Theorem 1, there exists a bounded net in such thatfor all . Since , set and and put in (9), we have and which is a contradiction. Thus, does not have an ultra central approximate identity.

Let and be Banach algebras with . Consider the Cartesian product . Equip with the norm and -Lau product, that is, the product which is defined byThen, becomes a Banach algebra which we denote it with . Note that (with the first Arens product) is isometrically isomorphism with , and also for every and in , we haveFor more information about the Lau product and its generalization, see [8–10].

Proposition 1. *Let**and**be Banach algebras with**. If**has an ultra central approximate identity, then**has an ultra central approximate identity.*

*Proof. *Let has an ultra central approximate identity. Then, there exists a net in such thatThus, we haveTherefore, for all . So, has an ultra central approximate identity.

Let be a Banach algebra and be a Banach -bimodule. The module extension is a Banach algebra with the following multiplication:and the norm . The Banach -bimodule is called commutative if for every and . It is easy to see that can be identified with (as a Banach space) and the first Arens product on is given byFor more information about the module extension, see [11].

Proposition 2. *Let**be a Banach algebra and**be a commutative Banach**-bimodule. If the module extension**has an ultra central approximate identity, then**has an ultra central approximate identity.*

*Proof. *Suppose that the module extension has an ultra central approximate identity. Then, there exists a net in such thatIt implies that for every and ,(i),(ii).So and . Thus, has an ultra central approximate identity.

#### 3. Applications for Certain Semigroup Algebras

We present some notions of the semigroup theory. The semigroup is called left cancellative, if for every in ,

The semigroup is called regular semigroup, if for every , there exists such that and [12].

Theorem 2. *Let**be a left cancellative regular semigroup. Then,**has an ultra central approximate identity if and only if**is a group.*

*Proof. *Suppose that has an ultra central approximate identity. Then, there exists a net in such that and for every . By Goldstein’s theorem, there exists a net in such that in . Thus, in for every . So,in . Applying iterated limit theorem [13], we have a net in such thatin . It is easy to see thatin . Let be a finite subset of . SetIt is easy to see that is a convex subset of , where for every , and . So, there exists a net (say again ) in whichfor every . Using a similar arguments as above, we may suppose that is a net in such thatfor every . So, has an approximate identity. It follows that . Then, there exists such that . Suppose in contradiction that for every , . Assume that has a form , where for every . Let . Since , there exists such that . Since , . So , which is a contradiction. Thus, for every , we have . The left cancellative property of implies that . Hence, is a left identity for . So,For every by equation (24), . On the other hand, is an approximate identity for . So, . It follows that , so is the identity of . Since is a regular, for every , there exists such thatAgain, the left cancellative property of implies that . It deduces that is a group.

For converse, if is a group, then has an identity. So has an identity. It follows that has an ultra central approximate identity.

A Banach algebra is contractible (or super amenable) if for every bounded derivation , there exists an element in such thatfor every Banach bimodule , see [1]. It is easy to see that every contractible Banach algebra is amenable.

Corollary 2. *Let**be a left cancellative semigroup. Then,**is pseudo-contractible if and only if**is a finite group.*

*Proof. *Let be a pseudo-contractible Banach algebra. So, we have a net in such that and for all . It is known that there is a bounded linear map such that for each and , we have(i),(ii), ,(iii),[[14], Lemma 1.]. Therefore, the net gives thatfor all . Then, we have satisfying to be an ultra central approximate identity for . Following a similar method as in [15], Theorem 1, and Corollary 2, we can see that is a regular semigroup. Thus, the previous theorem gives that must be a group, say . Then, is pseudo-contractible. On the other hand, is an algebra with identity, so possess an identity. Therefore [[2], Theorem 2.4] follows that is contractible. Then, is amenable. Now applying [[14], Theorem 1.3], is a finite group.

Converse is clear.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.