ω
                           
                           
                              ℒ
                           
                        
                     
                     
                  
               -biprojective and 
                  
                     
                     
                        
                           
                              ω
                           
                           
                              ¯
                           
                        
                     
                     
                  
               -contractible Banach algebras

Zahra Ghorbani

doi:10.1515/dema-2025-0118

Article Open Access

ω ℒ -biprojective and ω ¯ -contractible Banach algebras

Zahra Ghorbani

Published/Copyright: June 12, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

For a given Banach algebra ℳ and a continuous endomorphism ω on ℳ , we define ℳ to be ω ℒ -biprojective and ω ¯ -contractible. We then explore the relationship between them. Additionally, we show that l 1 ( N ∨ ) is a ω ℒ -biprojective Banach algebra. Finally, we examine the concept of ω -pseudo amenability and ω -approximate biprojectivity in Banach algebras. We demonstrate that for every unital Banach algebra ℳ , ω -approximate biprojectivity and ω -pseudo contractibility coincide.

Keywords: Banach algebra; ωℒ-biprojective; ω-contractible

MSC 2010: 43A20; 46H25; 47B47

1 Introduction

The concept of amenability was introduced by Johnson [1]. The author demonstrated that a Banach algebra ℳ is amenable if and only if it has a virtual diagonal. This means that if V ∈ ( ℳ ⊗ ℳ ˆ ) * * , then m ⋅ V = V ⋅ m for any m ∈ ℳ and ℱ ℳ * * ( V ) is an identity for ℳ , where ℱ ℳ : ( ℳ ⊗ ℳ ˆ ) ⟶ ℳ is defined by ℱ ℳ ( m ⊗ n ) = m n for every m , n ∈ ℳ , and is the morphism. The definition of a biprojective Banach algebra was initially provided by Helemskii [2,3]. According to this definition, a Banach algebra ℳ is biprojective if there is a bounded ℳ -bimodule map ϑ : ℳ ⟶ ℳ ⊗ ℳ ˆ such that ℱ ℳ ∘ ϑ = i d ℳ , where i d ℳ is the identity map on ℳ . It is known that every biprojective Banach algebra with a bounded approximate identity is amenable [3].

Let ω be a continuous endomorphism on ℳ . Mirzavaziri and Moslehian [4], Moslehian and Motlagh [5] described and investigated ω -amenability. A Banach algebra ℳ is called ω -amenable ( ω -contractible) if any ω -derivation from ℳ into K * (resp. K ) is ω -inner derivation, where K is a Banach ℳ -bimodule.

Ghorbani [6] introduced the notation of ω -biprojective as follows:

A Banach algebra ℳ is ω -biprojective, if there exists a ω - ℳ -bimodule homomorphism ϑ : ℳ ⟶ ( ℳ ⊗ ℳ ˆ ) such that ℱ ℳ ∘ ϑ = ω and further extended some results on biprojective Banach algebras. Yazdanpanah and Najafi [7] proved that if ω has a dense range, then ℳ is ω -contractible if and only if it is unital and ω -biprojective. For similar results on module versions of biprojectivity for Banach algebras, refer to [8–10].

Let ℳ be a Banach algebra and let ℒ be a closed ideal in ℳ . ℳ is ℒ -weakly amenable if every derivation from ℳ into ℒ * is an inner derivation. Furthermore, ℳ is ideally amenable if ℳ is ℒ -weakly amenable for every closed ideal ℒ in ℳ . This definition was introduced by Eshaghi Gordji and Yazdanpanah [11]. Rahimi and Tahmasebi introduced the notion of modulo amenability and contractibility; for more information, refer [12]. Rahimi and Soltani [13] introduced and studied the concept of approximate amenability (contractibility) modulo an ideal of Banach algebras. They also proved that if ℳ ∕ ℒ is approximately amenable and ℒ 2 = ℒ , then ℳ is approximately amenable modulo ℒ . Amenability modulo an ideal was also studied by Esmaili and Rahimi [14]. It was proven that being approximately contractible modulo an ideal, approximately amenable modulo an ideal, and w * -approximately amenable modulo an ideal in Banach algebras are equivalent. Teymouri et al. [15] introduced and examined ℳ ∕ ℒ -weak amenability and quotient ideal amenability for a Banach algebra ℳ . They established a relationship between these concepts and the weak amenability and ideal amenability of Banach algebras. It was demonstrated that if ℳ has a bounded approximate identity, then ℳ is ℳ ∕ ℒ -weakly amenable if and only if ℳ ∕ ℒ is weakly amenable. Khodakaramia et al. [16] studied ℒ -amenability. They explained that a Banach algebra ℳ is ℒ -amenable if ℳ ∕ ℒ is an amenable Banach algebra. They proved that if ℳ is ℒ -amenable and ℒ 2 ¯ = ℒ , then ℋ ℒ 1 ( ℳ , K * ) = { 0 } , for every Banach ℳ ∕ ℒ -bimodule K , where ℋ ℒ 1 ( ℳ , K * ) is the first cohomology group of ℳ with coefficients in K * .

In this study, we introduce and investigate ω ℒ -biprojective and ω ¯ -contractibility of Banach algebras. Additionally, we establish a relationship between the two definitions. We provide an example to show that the class of ω ℒ -biprojective Banach algebras is larger than that of biprojective Banach algebras. Let N ∨ be defined by the operation of the semigroup n 1 ∨ n 2 = max { n 1 , n 2 } , where n 1 , n 2 ∈ N . It is known that l 1 ( N ∨ ) with convolution is not biprojective ([17], Example 4.1.42). Finally, we demonstrate that if ω is a continuous endomorphism on l 1 ( N ∨ ) , then l 1 ( N ∨ ) is a ω ¯ -contractible and ω ℒ -biprojective Banach algebra if ω is a map with a dense range. It is important to note that if ω is a map with a dense range, then l 1 ( N ∨ ) is not ω -biprojective, since it is a unital Banach algebra. Furthermore, l 1 ( N ∨ ) is neither biprojective nor contractible Banach algebra [18].

2 Main results

Throughout this study, let ℳ be a Banach algebra, End ( ℳ ) be the set of all continuous endomorphisms on ℳ , and ℒ be a closed ideal of ℳ . We will use m ˆ to illustrate an element of ℳ ∕ ℒ . Additionally, we consider ω ∈ End ( ℳ ) as the map ω ˜ : ℳ ∕ ℒ ⟶ ℳ ∕ ℒ defined by ω ˜ ( m ˆ ) = ω ( m ) ^ and ω ( ℒ ) ⊂ ℒ . Let ℒ 0 denote the closed ideal defined by ℒ 0 = { m l : m ∈ ℳ , l ∈ ℒ } . The Banach algebra ℳ ∕ ℒ ⊗ ˆ ℳ ∕ ℒ 0 is a Banach ℳ -module with the following actions:

m 0 ⋅ ( m ˆ ⊗ n + ℒ 0 ) = m 0 m ^ ⊗ n + ℒ 0 , ( m ˆ ⊗ n + ℒ 0 ) ⋅ m 0 = m ˆ ⊗ n m 0 + ℒ 0 ( m , n , m 0 ∈ ℳ ) .

Consider ℱ : ℳ ∕ ℒ ⊗ ˆ ℳ ∕ ℒ 0 → ℳ ∕ ℒ defined by ℱ ( m ˆ ⊗ n + ℒ 0 ) = m n ^ for every m , n ∈ ℳ .

Definition 1

An element ν ∈ ( ℳ ∕ ℒ ⊗ ˆ ℳ ∕ ℒ 0 ) is said to be ω ˜ -diagonal for the Banach algebra ℳ if

ω ( m ) ⋅ ν = ν ⋅ ω ( m ) ,
ℱ ( ν ) ⋅ ω ( m ) = ω ˜ ( m ˆ ) , ( m ∈ ℳ ) .

If ν is a ω ˜ -diagonal for ℳ , then ℱ ( ν ⋅ ω ( m ) ) = ℱ ( ν ) ⋅ ω ˜ ( m ˆ ) = ω ˜ ( m ˆ ) in which ℱ ( ν ) is a unit for ω ˜ ( ℳ ∕ ℒ ) . Also, if ω is a map with dense range, then ℱ ( ν ) = e ˆ is a unit for ℳ ∕ ℒ .

Proposition 1

Let ℳ have a ω ˜ -diagonal. Then, ℳ has a ω ∘ ψ ˜ -diagonal for every ψ ∈ End ( ℳ ) .

Proof

Suppose that ν ∈ ℳ ∕ ℒ ⊗ ˆ ℳ ∕ ℒ 0 is a ω ˜ -diagonal for ℳ . Therefore, we obtain ω ( m ) ⋅ ν = ν ⋅ ω ( m ) and ℱ ( ν ) ⋅ ω ( m ) = ω ˜ ( m ˆ ) , ( m ∈ ℳ ) . So, ω ( ψ ( m ) ) ⋅ ν = ν ⋅ ω ( ψ ( m ) ) and ℱ ( ν ) ⋅ ω ( ψ ( m ) ) = ω ∘ ψ ˜ ( m ˆ ) for all m ∈ ℳ , i.e., ν is a ω ∘ ψ ˜ -diagonal for ℳ .□

Proposition 2

Let ℳ and N be Banach algebras and K be a closed ideal of N . Let ψ ∈ End ( N ) , if ℳ has a ω ˜ -diagonal and N has a ψ ˜ -diagonal, then ℳ ⊗ N ˆ has a ω ⊗ ψ ˜ -diagonal.

Proof

Assume that K 0 is the closed ideal { n k : n ∈ N , k ∈ K } . Let ν ℳ = Σ i = 1 ∞ m i ˆ ⊗ ( m ́ i + ℒ 0 ) be a ω ˜ -diagonal for ℳ and ν N = Σ j = 1 ∞ ( n j + K ) ⊗ ( ń j + K 0 ) be a ψ ˜ -diagonal for N . We define

ν ≔ Σ i , j = 1 ∞ ( m i ˆ ⊗ ( n j + K ) ) ⊗ ( ( m ́ i + ℒ 0 ) ⊗ ( ń j + K 0 ) ) .

Then, for every m ∈ ℳ and n ∈ N , we obtain

□ ℱ ( ν ) ⋅ ( ω ⊗ ψ ) ( m ⊗ n ) = Σ i , i = 1 ∞ ( m i ˆ ⊗ ( n j + K ) ) ⋅ ( ( m ́ i + ℒ 0 ) ⊗ ( ń j + K 0 ) ) ⋅ ( ω ( m ) ⊗ ψ ( n ) ) = Σ i = 1 ∞ m i ˆ ( m ́ i + ℒ 0 ) ω ( m ) ⊗ Σ j = 1 ∞ ( n j + K ) ( ń j + K 0 ) ψ ( n ) = ℱ ( ν ℳ ) ⋅ ω ( m ) ⊗ ℱ ( ν N ) ψ ( n ) = ω ˜ ( m ˆ ) ⊗ ψ ˜ ( n + K ) .

Suppose K and N are Banach ℳ -bimodules. A ω - ℳ -bimodule morphism ϑ : K ⟶ N is a morphism such that for any m ∈ ℳ and k ∈ K , we have

ϑ ( m ⋅ k ) = ω ( m ) ⋅ ϑ ( k ) , ϑ ( k ⋅ m ) = ϑ ( k ) ⋅ ω ( m ) .

Definition 2

A Banach algebra ℳ is considered to be ω ℒ -biprojective if there is a ω - ℳ -bimodule homomorphism ϑ : ℳ ∕ ℒ ⟶ ℳ ∕ ℒ ⊗ ˆ ℳ ∕ ℒ 0 with ℱ ∘ ϑ = ω ˜ .

Theorem 1

A unital Banach algebra ℳ is ω ℒ -biprojective if and only if it has a ω ˜ -diagonal.

Proof

Let ν be a ω ˜ -diagonal for ℳ . We define ϑ : ℳ ∕ ℒ ⟶ ℳ ∕ ℒ ⊗ ˆ ℳ ∕ ℒ 0 by ϑ ( m ˆ ) = ω ( m ) ⋅ ν . Since

ω ( m n ) ⋅ ν = ω ( m ) ω ( n ) ⋅ ν = ω ( m ) ⋅ ν ⋅ ω ( n ) , ( m , n ∈ ℳ ) ,

we have ϑ ( m n ^ ) = ω ( m n ) ⋅ ν = ω ( m ) ω ( n ) ⋅ ν = ω ( m ) ⋅ ϑ ( n ˆ ) and ϑ ( n m ^ ) = ω ( n m ) ⋅ ν = ω ( n ) ⋅ ν ⋅ ω ( m ) = ϑ ( n ˆ ) ⋅ ω ( m ) . It follows that ϑ is a ω - ℳ -bimodule homomorphism. Furthermore,

ℱ ∘ ϑ ( m ˆ ) = ℱ ( ω ( m ) ⋅ ν ) = ℱ ( ν ⋅ ω ( m ) ) = ℱ ( ν ) ⋅ ω ( m ) = ω ˜ ( m ˆ ) .

Therefore, ℳ is a ω ℒ -biprojective Banach algebra.

Conversely, if e is an identity for ℳ and ϑ : ℳ ∕ ℒ ⟶ ℳ ∕ ℒ ⊗ ˆ ℳ ∕ ℒ 0 is a ω - ℳ -bimodule homomorphism with ℱ ∘ ϑ = ω ˜ . If ν = ϑ ( e ˆ ) , for any m ∈ ℳ , we have

ω ( m ) ⋅ ν = ω ( m ) ⋅ ϑ ( e ˆ ) = ϑ ( m ⋅ e ˆ ) = ϑ ( m e ^ ) = ϑ ( e m ^ ) = ϑ ( e ˆ ⋅ m ) = ϑ ( e ˆ ) ⋅ ω ( m ) = ν ⋅ ω ( m ) ,

and

ℱ ( ν ) ⋅ ω ( m ) = ℱ ( ϑ ( e ˆ ) ) ⋅ ω ( m ) = ω ˜ ( e ˆ ) ⋅ ω ( m ) = ω ( e ) ^ ⋅ ω ( m ) = ω ( e m ) ^ = ω ( m ) ^ = ω ˜ ( m ˆ ) .

Then, ν is a ω ˜ -diagonal for ℳ .□

Proposition 3

If ℳ is ω ℒ -biprojective, then ℳ ∕ ℒ is ω ˜ -biprojective.

Proof

Given that ℳ is ω ℒ -biprojective, there exists a bounded ω - ℳ -bimodule homomorphism ϑ : ℳ ∕ ℒ ⟶ ℳ ∕ ℒ ⊗ ˆ ℳ ∕ ℒ 0 with ℱ ∘ ϑ = ω ˜ . Let ϑ ( m ˆ ) = Σ i = 1 ∞ ( m i ˆ ) ⊗ ( n i + ℒ 0 ) , where ( m i ˆ ) i ∈ ℳ ∕ ℒ , ( n i + ℒ 0 ) i ∈ ℳ ∕ ℒ 0 . Define ℳ -bimodule homomorphism ι : ℳ ∕ ℒ 0 → ℳ ∕ ℒ by ι ( m + ℒ 0 ) = m ˆ . Put ϑ ˜ = ( i d ℳ ∕ ℒ ⊗ ι ) ∘ ϑ . Since ℒ ⋅ ( ℳ ∕ ℒ ) = ( ℳ ∕ ℒ ) ⋅ ℒ = 0 , ϑ ˜ is a ω ˜ - ℳ ∕ ℒ -bimodule homomorphism and we have

ℱ ℳ ∕ ℒ ∘ ϑ ˜ ( m ˆ ) = ℱ ℳ ∕ ℒ ∘ ( ( i d ℳ ∕ ℒ ⊗ ι ) ∘ ϑ ) ( m ˆ ) = ℱ ℳ ∕ ℒ ∘ ( i d ℳ ∕ ℒ ⊗ ι ) ( Σ i = 1 ∞ ( m i ˆ ) ⊗ ( n i + ℒ 0 ) ) = ℱ ℳ ∕ ℒ ( Σ i = 1 ∞ ( m i ˆ ) ⊗ ( n i ˆ ) ) = Σ i = 1 ∞ ( m i ˆ ) ( n i ˆ ) = Σ i = 1 ∞ ( m i n i ^ ) = ℱ ( Σ i = 1 ∞ ( m i ˆ ) ⊗ ( n i + ℒ 0 ) ) = ℱ ∘ ϑ ( m ˆ ) = ω ˜ ( m ˆ ) .

Therefore, ℳ ∕ ℒ is ω ˜ -biprojective.□

For any Banach ℳ -bimodule K with K ⋅ ℒ = ℒ ⋅ K = 0 and a ω -derivation from ℳ into K * ( K ) , which is a ω -inner derivation on ℳ − ℒ , we call ℳ is ω ¯ -amenable ( ω ¯ -contractible).

Proposition 4

If ℳ is ω ¯ -contractible, then for every ψ ∈ End ( ℳ ) , ℳ is ψ ∘ ω ¯ -contractible.

Proof

Let D : ℳ ⟶ K be a ψ ∘ ω -derivation with ℒ ⋅ K = K ⋅ ℒ = 0 . Then, K is an ℳ -bimodule via

m ⋅ k = ψ ( m ) ⋅ k , k ⋅ m = k ⋅ ψ ( m ) ( m ∈ ℳ , k ∈ K ) .

For every m , n ∈ ℳ , we have

D ( m n ) = ψ ∘ ω ( m ) ⋅ D ( n ) + D ( m ) ⋅ ψ ∘ ω ( n ) = ω ( m ) ⋅ D ( n ) + D ( m ) ⋅ ω ( n ) .

Hence, D is a ω -derivation. By the ω ¯ -contractibility of ℳ , there is k ∈ K such that

□ D ( m ) = ω ( m ) ⋅ k − k ⋅ ω ( m ) = ψ ∘ ω ( m ) ⋅ k − k ⋅ ψ ∘ ω ( m ) ( m ∈ ℳ − ℒ ) .

Proposition 5

The following statements are valid:

Let ℳ ∕ ℒ be ω ˜ -contractible and ℒ is an idempotent, then ℳ is ω ¯ -contractible.
Let ℳ be ω ¯ -contractible, then ℳ ∕ ℒ is ω ˜ -contractible.
Assuming ℳ is ω ¯ -contractible, ω is an idempotent homomorphism, and ℒ is ω ∣ ℒ -contractible, then ℳ is ω -contractible.

Proof

(i) Let D : ℳ ⟶ K be a ω -derivation and ℒ ⋅ K = K ⋅ ℒ = 0 . Given D ˜ : ℳ ∕ ℒ ⟶ K by D ˜ ( m ˆ ) = D ( m ) . Since ℒ is idempotent, for every l ∈ ℒ , D ( l ) = D ( l 2 ) = ω ( l ) ⋅ D ( l ) + D ( l ) ⋅ ω ( l ) = 0 . Hence, D ˜ is well-defined. By the ω ˜ -contractibility of ℳ ∕ ℒ , there is a k ∈ K such that D ( m ) = D ˜ ( m ˆ ) = ω ˜ ( m ) ⋅ k − k ⋅ ω ˜ ( m ) = ω ( m ) ^ ⋅ k − k ⋅ ω ( m ) ^ = ω ( m ) ⋅ k − k ⋅ ω ( m ) .

(ii) Assume ℳ is ω ¯ -contractible and D : ℳ ∕ ℒ ⟶ K is a ω ˜ -derivation. Then, K is an ℳ -bimodule via:

m ⋅ k = q ( m ) ⋅ k , k ⋅ m = k ⋅ q ( m ) , ( m ∈ ℳ , k ∈ K ) ,

where q : ℳ ⟶ ℳ ∕ ℒ by m ↦ m ˆ . Then, we obtain ℒ ⋅ K = K ⋅ ℒ = 0 and

D ∘ q ( m n ) = D ( m n ^ ) = ω ˜ ( m ) ⋅ D ( n ˆ ) + D ( m ˆ ) ⋅ ω ˜ ( n ) = ω ( m ) ^ ⋅ D ( n ˆ ) + D ( m ˆ ) ⋅ ω ( n ) ^ = ω ( m ) ⋅ D ( n ˆ ) + D ( m ˆ ) ⋅ ω ( n ) = ω ( m ) ⋅ D ∘ q ( n ) + D ∘ q ( m ) ⋅ ω ( n ) .

Therefore, D ∘ q is a ω -derivation on ℳ . Hence, there is k ∈ K with

D ∘ q ( m ) = ω ( m ) ⋅ k − k ⋅ ω ( m ) .

Thus,

D ( m ˆ ) = D ∘ q ( m ) = ω ( m ) ⋅ k − k ⋅ ω ( m ) = ω ( m ) ^ ⋅ k − k ⋅ ω ( m ) ^ = ω ˜ ( m ) ⋅ k − k ⋅ ω ˜ ( m ) .

Therefore, ℳ ∕ ℒ is ω ˜ -contractible.

(iii) Let ℳ be ω ¯ -contractible, similar to (ii), ℳ ∕ ℒ is ω ˜ -contractible. Since ℒ is ω ∣ ℒ -contractible, it follows from proposition 3.2 [5] that ℳ is ω -contractible.□

Theorem 2

Suppose that ω has a dense range. Then, ℳ ∕ ℒ is a ω ˜ -contractible if and only if ℳ has a ω ˜ -diagonal.

Proof

Let D : ℳ ∕ ℒ ⟶ K be a ω ˜ -derivation. We define T : ℳ ∕ ℒ ⊗ ˆ ℳ ∕ ℒ 0 ⟶ K by T ( m ˆ ⊗ ( n + ℒ 0 ) ) ≔ ω ˜ ( m ˆ ) ⋅ D ( n ˆ ) . Assume that ν = Σ i = 1 ∞ ( m i ˆ ) ⊗ ( n i + ℒ 0 ) with Σ i = 1 ∞ ‖ m i ‖ ‖ n i ‖ < ∞ and a ω ˜ -diagonal for ℳ . Therefore, ℱ ( ν ) = Σ i = 1 ∞ ( m i n i ^ ) = e ˆ is a unit for ℳ ∕ ℒ . Since ω is a map with a dense range, for all m ∈ ℳ , we have T ( m ⋅ ν ) = T ( ν ⋅ m ) . Hence, we have

Σ i = 1 ∞ ω ˜ ( m m i ^ ) ⋅ D ( n i ˆ ) = Σ i = 1 ∞ ω ˜ ( m i ˆ ) ⋅ D ( n i m ^ ) ( m ∈ ℳ ) .

For k = Σ i = 1 ∞ ω ˜ ( m i ˆ ) D ( n i ˆ ) + ω ˜ ( e ˆ ) ⋅ D ( e ˆ ) − D ( e ˆ ) , k ∈ K . Then, for every m ˆ ∈ ℳ ∕ ℒ , we obtain

ω ˜ ( m ˆ ) ⋅ k − k ⋅ ω ˜ ( m ˆ ) = Σ i = 1 ∞ ω ˜ ( m m i ^ ) ⋅ D ( n i ˆ ) − Σ i = 1 ∞ ω ˜ ( m i ˆ ) ⋅ D ( n i ˆ ) ⋅ ω ˜ ( m ˆ ) + ω ˜ ( m ˆ ) ⋅ D ( e ˆ ) − ω ˜ ( m ˆ ) ⋅ D ( e ˆ ) − ω ˜ ( e ˆ ) ⋅ D ( e ˆ ) ⋅ ω ˜ ( m ˆ ) + D ( e ˆ ) ⋅ ω ˜ ( m ˆ ) = Σ i = 1 ∞ ω ˜ ( m i ˆ ) ⋅ D ( n i m ^ ) − Σ i = 1 ∞ ω ˜ ( m i ˆ ) ⋅ D ( n i ˆ ) ⋅ ω ˜ ( m ˆ ) − ω ˜ ( e ˆ ) ⋅ D ( e ˆ ) ⋅ ω ˜ ( m ˆ ) + D ( e ˆ ) ⋅ ω ˜ ( m ˆ ) = Σ i = 1 ∞ ω ˜ ( m i ˆ ) ⋅ D ( n i ^ ) ⋅ ω ˜ ( m ˆ ) + Σ i = 1 ∞ ω ˜ ( m i ˆ ) ω ˜ ( n i ˆ ) ⋅ D ( m ^ ) − Σ i = 1 ∞ ω ˜ ( m i ˆ ) ⋅ D ( n i ˆ ) ⋅ ω ˜ ( m ˆ ) − ω ˜ ( e ˆ ) ⋅ D ( e ˆ ) ⋅ ω ˜ ( m ˆ ) + D ( e ˆ ) ⋅ ω ˜ ( m ˆ ) = Σ i = 1 ∞ ω ˜ ( m i n i ^ ) ⋅ D ( m ˆ ) − ω ˜ ( e ˆ ) ⋅ D ( e ˆ ) ⋅ ω ˜ ( m ˆ ) + D ( e ˆ ) ⋅ ω ˜ ( m ˆ ) = ω ˜ ( ℱ ( ν ) ) ⋅ D ( m ˆ ) − ω ˜ ( e ˆ ) ⋅ D ( e ˆ ) ⋅ ω ˜ ( m ˆ ) + D ( e ˆ ) ⋅ ω ˜ ( m ˆ ) + ω ˜ ( e ˆ ) ⋅ D ( m ˆ ) − ω ˜ ( e ˆ ) ⋅ D ( m ˆ ) = ω ˜ ( e ˆ ) ⋅ D ( m ˆ ) + [ D ( e ˆ ) ⋅ ω ˜ ( m ˆ ) + ω ˜ ( e ˆ ) ⋅ D ( m ˆ ) ] − ω ˜ ( e ˆ ) ⋅ [ ω ˜ ( e ˆ ) ⋅ D ( m ˆ ) + D ( e ˆ ) ⋅ ω ˜ ( m ˆ ) ] = ω ˜ ( e ˆ ) ⋅ D ( m ˆ ) + D ( e m ^ ) − ω ˜ ( e ˆ ) ⋅ D ( e m ^ ) = ω ˜ ( e ˆ ) ⋅ D ( m ˆ ) + D ( m ˆ ) − ω ˜ ( e ˆ ) ⋅ D ( m ˆ ) = D ( m ˆ ) .

Hence, D is a ω ˜ -inner derivation. Conversely, if ℳ ∕ ℒ is ω ˜ -contractible, then by Corollary 3.2 [7], ℳ ∕ ℒ has a unit e ˆ . Let K = ℳ ∕ ℒ ⊗ ˆ ℳ ∕ ℒ 0 . Since ℒ ⋅ K = K ⋅ ℒ = 0 , we assume K is a ℳ -bimodule as a ℳ ∕ ℒ -bimodule Banach via

m ˆ ⋅ k = m ⋅ k , k ⋅ m ˆ = k ⋅ m ( m ∈ ℳ , k ∈ K ) .

For every m ˆ ∈ ℳ ∕ ℒ , we define D : ℳ ∕ ℒ ⟶ ker ℱ by

D ( m ˆ ) = ω ˜ ( m ˆ ) ⋅ ( e ˆ ⊗ e ˆ ) − ( e ˆ ⊗ e ˆ ) ⋅ ω ˜ ( m ˆ ) .

Therefore, D is a ω ˜ -derivation. Since ℳ ∕ ℒ is a ω ˜ -contractible, there exists a n ∈ ker ℱ with D ( m ˆ ) = ω ˜ ( m ˆ ) ⋅ n − n ⋅ ω ˜ ( m ˆ ) . Take ν = e ˆ ⊗ e ˆ − n . Since D ( m ˆ ) = ω ˜ ( m ˆ ) ⋅ ( e ˆ ⊗ e ˆ ) − ( e ˆ ⊗ e ˆ ) ⋅ ω ˜ ( m ˆ ) = ω ˜ ( m ˆ ) ⋅ n − n ⋅ ω ˜ ( m ˆ ) , then we have

ω ˜ ( m ˆ ) ⋅ ( e ˆ ⊗ e ˆ ) − ω ˜ ( m ˆ ) ⋅ n = ( e ˆ ⊗ e ˆ ) ⋅ ω ˜ ( m ˆ ) − n ⋅ ω ˜ ( m ˆ ) .

Therefore, ω ( m ) ⋅ ν = ν ⋅ ω ( m ) and ℱ ( ν ) ⋅ ω ( m ) = ℱ ( e ˆ ⊗ e ˆ − n ) ⋅ ω ( m ) = ω ( m ) ^ = ω ˜ ( m ˆ ) . Thus, ν is a ω ˜ -diagonal for ℳ .□

Theorem 3

Let ℒ be an idempotent ideal of ℳ and ω be a map with a dense range. Then, ℳ is a ω ¯ -contractible if and only if it has a ω ˜ -diagonal.

Proof

Suppose that ℳ be a ω ¯ -contractible. Then, by Proposition 5, ℳ ∕ ℒ is a ω ˜ -contractible. Therefore, by Theorem 2, ℳ has a ω ˜ -diagonal. Conversely, if ℳ has a ω ˜ -diagonal, then by Theorem 2, ℳ ∕ ℒ is ω ˜ -contractible and so by Proposition 5, ℳ is a ω ¯ -contractible.□

Example 1

Let S = { b i d j : i , j ≥ 0 } be the bicyclic inverse semigroup generated by b and d . Let E be the set of idempotents in S , then E = { b i d i : i = 0 , 1 , … } . We define equivalence on S as follows:

b ∼ d ⇔ ∃ c ∈ E : c b = c d , ( b , d ∈ S ) .

Munn [19] showed that S ∕ ∼ is homomorphic to the maximal group homomorphic image G S of S . Therefore, S ∕ ∼ = G S = Z , hence S ∕ ∼ is amenable. By Johnson’s theorem [1], l 1 ( S ∕ ∼ ) is amenable. Define ϑ : S → S ∕ ∼ , which extends to an epimorphism ϑ ˜ : l 1 ( S ) → l 1 ( S ∕ ∼ ) [20]. Therefore, l 1 ( S ) ∕ ker ϑ ˜ ≃ l 1 ( S ∕ ∼ ) . Let ω ∈ End ( l 1 ( S ) ) , we define ω ˜ : l 1 ( S ) ∕ ker ϑ ˜ ⟶ l 1 ( S ) ∕ ker ϑ ˜ by ω ˜ ( δ s + ℒ ) = ω ( δ s ) + ℒ , where ℒ = ker ϑ ˜ . Since l 1 ( S ∕ ∼ ) is amenable, therefore l 1 ( S ) ∕ ker ϑ ˜ is amenable, hence it is ω ˜ -amenable. If ω is a dense range map, then l 1 ( S ) ∕ ker ϑ ˜ , is not ω ˜ -contractible, otherwise l 1 ( S ) ∕ ker ϑ ˜ ≃ l 1 ( S ∕ ∼ ) would be contractible, which is a contradiction because S ∕ ∼ is infinite [21]. Therefore, by Proposition 5, l 1 ( S ) is not ω ˜ -contractible, where ℒ = ker ϑ ˜ .

Proposition 6

Let ℳ be a ω ¯ -contractible unital Banach algebra. If ω is a map with a dense range, then ℳ is ω ℒ -biprojective.

Proof

Since ℳ is a ω ¯ -contractible, by Proposition 5 (ii), ℳ ∕ ℒ is ω ˜ -contractible and so, by Theorem 2, ℳ has a ω ˜ -diagonal. Hence, by Theorem 1, ℳ is a ω ℒ -biprojective Banach algebra.□

Proposition 7

Let ℳ be a ω ℒ -biprojective unital Banach algebra and ℒ be an idempotent ideal. Then, ℳ is a ω ¯ -contractible.

Proof

Assume that ℳ is a ω ℒ -biprojective unital Banach algebra. By Theorem 1, ℳ has a ω ˜ -diagonal and so by Theorem 2, ℳ ∕ ℒ is a ω ˜ -contractible. Since ℒ is an idempotent ideal, by Proposition 5 (i), ℳ is a ω ¯ -contractible.□

Proposition 8

Let ℳ ∕ ℒ be a biprojective unital Banach algebra. If ω is a map with a dense range, then ℳ is a ω ℒ -biprojective.

Proof

Since ℳ ∕ ℒ is a biprojective unital Banach algebra, by Theorem 2.8.48 [17], ℳ ∕ ℒ is contractible. Thus for every ω ∈ End ( ℳ ) , ℳ ∕ ℒ is a ω ˜ -contractible. So by Theorem 2, ℳ has a ω ˜ -diagonal. Using Theorem 1, ℳ is a ω ℒ -biprojective.□

Example 2

Let S = N ∨ be defined by the operation of the semigroup n 1 ∨ n 2 = max { n 1 , n 2 } , where n 1 , n 2 ∈ N . It is known that l 1 ( N ∨ ) with convolution is not biprojective [17], Example 4.1.42. Let E be the set of idempotents on N ∨ , then E = N . Introduce equivalence on N ∨ as follows:

a ∼ b ⇔ ∃ c ∈ E : c a = c b , ( a , b ∈ N ∨ ) .

Define ϑ : N ∨ → N ∨ ∕ ∼ , which extends to an epimorphism ϑ ˜ : l 1 ( N ∨ ) → l 1 ( N ∨ ∕ ∼ ) [20]. Therefore, l 1 ( N ∨ ) ∕ ker ϑ ˜ ≃ l 1 ( N ∨ ∕ ∼ ) . Munn [19] showed that S ∕ ∼ is homomorphic to the maximal group homomorphic image G S of S . Selivanov [21] demonstrated that S ∕ ∼ is finite if and only if l 1 ( S ∕ ∼ ) is contractible. Since G S is finite, l 1 ( N ∨ ) ∕ ker ϑ ˜ is contractible, and thus for every ω ∈ End ( l 1 ( N ∨ ) ) , l 1 ( N ∨ ) ∕ ker ϑ ˜ , ω ˜ -contractible. Following from Proposition 5, l 1 ( N ∨ ) is ω ¯ -contractible, and it is known that δ 1 is the identity for l 1 ( N ∨ ) . If ω is a map with a dense range, by Proposition 6, l 1 ( N ∨ ) is ω ℒ -biprojective, where ℒ = ker ϑ ˜ . Note that l 1 ( N ∨ ) is not contractible.

We are reminded that ℳ is ω -approximate biprojective if there exists a continuous ω - ℳ -bimodule homomorphism ϑ α : ℳ ⟶ ( ℳ ⊗ ℳ ˆ ) with ℱ ℳ ∘ ϑ α ( m ) → ω ( m ) [23]. A bounded ω -approximate diagonal for ℳ is a bounded net ( ν α ) in ( ℳ ⊗ ℳ ˆ ) , which ν α ⋅ ω ( m ) − ω ( m ) ⋅ ν α ⟶ 0 and ℱ ℳ ( ν α ) ⋅ ω ( m ) ⟶ ω ( m ) for any m ∈ ℳ [24].

Proposition 9

Let ℳ be a ω -approximate biprojective with the central element m 0 such that ω ( m m 0 ) = ω ( m ) . Then, ℳ has a ω -approximate diagonal.

Proof

Suppose ℳ is a ω -approximate biprojective. There exists a net ϑ α : ℳ ⟶ ( ℳ ⊗ ℳ ˆ ) of ω - ℳ -bimodule homomorphisms such that ℱ ℳ ∘ ϑ α ( m ) → ω ( m ) . Take ν α = ϑ α ( m 0 ) , then for any m ∈ ℳ , we have

ν α ⋅ ω ( m ) − ω ( m ) ⋅ ν α = ϑ α ( m 0 ) ⋅ ω ( m ) − ω ( m ) ⋅ ϑ α ( m 0 ) = ϑ α ( m 0 m ) − ϑ α ( m m 0 ) ⟶ 0 ,

and so, ℱ ℳ ( ν α ) ⋅ ω ( m ) = ℱ ℳ ( ϑ α ( m 0 ) ) ⋅ ω ( m ) → ω ( m 0 m ) = ω ( m ) . Therefore, ν α is ω -approximate diagonal for ℳ .□

Suppose that any ω -derivation from ℳ into K * is a ω -approximate inner derivation, meaning there is a net ( k α ) ⊂ K * such that D ( m ) = lim α ω ( m ) ⋅ k α − k α ⋅ ω ( k ) , where m ∈ ℳ . In this case, we say that ℳ is ω -approximate amenable [23].

Corollary 1

Let ℳ be a ω -approximate biprojective with the central element m 0 such that ω ( m m 0 ) = ω ( m ) and ω is an epimorphism. Then, ℳ is ω -approximately amenable.

Proof

Assume that ℳ be a ω -approximate biprojective, by Proposition 9, ℳ has a ω -approximate diagonal. By Theorem 2.6 [24], ℳ is ω -approximately amenable.□

Corollary 2

Let ℳ be a ω -approximate biprojective with a central bounded approximate identity and ω be an epimorphism. Then, ℳ is ω -approximately amenable.

If there is a net ( ν α ) ⊂ ℳ ⊗ ℳ ˆ such that lim α ( ν α ⋅ ω ( m ) − ω ( m ) ⋅ ν α ) = 0 and lim α ℱ ℳ ( ν α ) ⋅ ω ( m ) = ω ( m ) , for m ∈ ℳ , then we say that ℳ is ω -pseudo amenable.

Proposition 10

If ℳ is ω -pseudo amenable, then ℳ is ω -approximate biprojective.

Proof

Let ℳ be ω -pseudo amenable. Then, there is a net ( ν α ) ⊂ ℳ ⊗ ℳ ˆ with lim α ( ν α ⋅ ω ( m ) − ω ( m ) ⋅ ν α ) = 0 and lim α ℱ ℳ ( ν α ) ⋅ ω ( m ) = ω ( m ) , for m ∈ ℳ . Take ϑ α ( m ) = ω ( m ) ⋅ ν α , then for every m ∈ ℳ , we have

□ ℱ ℳ ∘ ϑ α ( m ) = ℱ ℳ ( ω ( m ) ⋅ ν α ) = ω ( m ) ⋅ ℱ ℳ ( ν α ) → ω ( m ) .

Example 3

Consider ℳ = m n 0 0 : m , n ∈ C under the l 1 norm. Then, ℳ is a Banach algebra. If

m 0 = 1 0 0 0 ,

then m 0 is the central element of ℳ . Define

ϑ α m n 0 0 = m ( m 0 ⊗ m 0 ) .

Then, for m ∈ ℳ and ω m n 0 0 = m 0 0 0 , we see that ϑ α is a ω - ℳ -bimodule homomorphism and ℱ ℳ ∘ ϑ α → ω . Thus, ℳ is ω -approximate biprojective Banach algebra. Since ω m n 0 0 m 0 = ω m n 0 0 by Proposition 9, ℳ has a ω -approximate diagonal.

Proposition 11

If ℳ has an ω -approximate diagonal, then ℳ is ω -approximate biprojective.

Proof

Let ( ν α ) in ( ℳ ⊗ ℳ ˆ ) be an ω -approximate diagonal. We define ϑ α : ℳ ⟶ ( ℳ ⊗ ℳ ˆ ) by m ↦ ω ( m ) ⋅ ν α ( m ∈ ℳ ). Then, for any m ∈ ℳ , we obtain

□ ℱ ℳ ∘ ϑ α ( m ) = ℱ ℳ ( ω ( m ) ⋅ ν α ) = ω ( m ) ⋅ ℱ ℳ ( ν α ) ⟶ ω ( m ) .

A Banach algebra ℳ is biflat if there is a bounded ℳ -bimodule map ϑ from ( ℳ ⊗ ℳ ˆ ) * * into ℳ * with ϑ ∘ ℱ ℳ * an identity map on ℳ * [2]. Ghorbani [6] introduced the notation of ω -Helemskii biflat. A Banach algebra ℳ is said to be ω -Helemskii biflat, if there is a ω - ℳ -bimodule homomorphism ϑ from ℳ into ( ℳ ⊗ ℳ ˆ ) * * such that ℱ ℳ * * ∘ ϑ = κ ℳ ∘ ω , where κ ℳ is the natural embedding map of ℳ into ℳ * * and then they generalized some results on biflat Banach algebras.

Theorem 4

Let ℳ be a biflat, then ℳ is ω -approximate biprojective.

Proof

Suppose ℳ is a biflat, then ℳ is a ω -Helemskii biflat [6]. Thus, there is a ω - ℳ -bimodule morphisms ϑ : ℳ ⟶ ( ℳ ⊗ ℳ ˆ ) * * with ℱ ℳ * * ∘ ϑ = κ ℳ ∘ ω . This leads to a net ϑ α : ℳ ⟶ ( ℳ ⊗ ℳ ˆ ) , ( α ∈ ▵ ) such that ϑ = W * − lim α ϑ α . So, for any m ∈ ℳ , we have

W * − lim α ℱ ℳ ∘ ϑ α ( m ) = W * − lim α ℱ ℳ * * ∘ ϑ α ( m ) = ℱ ℳ * * ∘ ϑ ( m ) = ω ( m ) .

Hence, there exists a net ϑ α ˜ in B ( ℳ , ℳ ⊗ ℳ ˆ ) [22] such that

□ ℱ ℳ ∘ ϑ α ˜ ( m ) → ω ( m ) .

Recall that ℳ is ω -pseudo contractible if there is a ω -approximate diagonal ( ν α ) ⊂ ℳ ⊗ ℳ ˆ such that ω ( m ) ⋅ ν α = ν α ⋅ ω ( m ) for all m ∈ ℳ and all α [23].

Theorem 5

Suppose that ℳ be ω -approximate biprojective with unital. Then, ℳ is ω -pseudo contractible.

Proof

Let ℳ be ω -approximate biprojective, implying a net ϑ α : ℳ ⟶ ( ℳ ⊗ ℳ ˆ ) , ( α ∈ ▵ ) of ω - ℳ -bimodule homomorphisms with lim α ℱ ℳ ∘ ϑ α ( m ) = ω ( m ) , ( m ∈ ℳ ) . Let e ℳ be the unit of ℳ and define ν α = ϑ α ( e ℳ ) , for every m ∈ ℳ . Then, we have

ω ( m ) ⋅ ν α = ω ( m ) ⋅ ϑ α ( e ℳ ) = ϑ α ( m e ℳ ) = ϑ α ( e ℳ m ) = ϑ α ( e ℳ ) ⋅ ω ( m ) = ν α ⋅ ω ( m ) .

Also, for each m ∈ ℳ , we obtain

ℱ ℳ ( ν α ) ⋅ ω ( m ) = ℱ ℳ ( ϑ α ( e ℳ ) ) ⋅ ω ( m ) → ω ( m ) .

Hence, ℳ is ω -pseudo contractible.□

Proposition 12

If ℳ is ω -pseudo contractible, then ℳ is ω -approximate biprojective.

Proof

Assuming ℳ is ω -pseudo contractible, therefore there is a net ( ν α ) ⊂ ℳ ⊗ ℳ ˆ such that ω -approximate diagonal for ℳ and ω ( m ) ⋅ ν α = ν α ⋅ ω ( m ) for all m ∈ ℳ . We define ϑ α : ℳ ⟶ ( ℳ ⊗ ℳ ˆ ) by ϑ α ( m ) ≔ ω ( m ) ⋅ ν α . Then, for any m ∈ ℳ , we have

□ lim α ℱ ℳ ∘ ϑ α ( m ) = lim α ℱ ℳ ( ω ( m ) ⋅ ν α ) = ω ( m ) .

Definition 3

Given Banach algebra ℳ with the norm ‖ . ‖ ℳ , a Banach algebra K with the norm ‖ . ‖ K is an abstract Segal algebra with respect to ℳ if the following statements hold:

K ¯ = ℳ where K is the left ideal in ℳ .
There exists C > 0 with ‖ k ‖ ℳ ≤ C ‖ k ‖ K , ( k ∈ K ) .
There exists l > 0 with ‖ k 1 k 2 ‖ K ≤ l ‖ k 1 ‖ ℳ ‖ k 2 ‖ K , ( k 1 , k 2 ∈ K ) .

Theorem 6

Suppose that K be an abstract Segal algebra with an approximate identity with respect to ℳ . If ω ( K ) = K , then K is ω ∣ K -biprojective if and only if K is ω ∣ K -contractible.

Proof

If K is ω ∣ K -contractible, then by Theorem 2.2 [24], K has a ω ∣ K -diagonal ν . We define ϑ : K ⟶ ( K ⊗ K ˆ ) by k ↦ ω ∣ K ( k ) ⋅ ν . For every k ∈ K , we have

ℱ K ∘ ϑ ( k ) = ℱ K ∘ ( ω ∣ K ( k ) ⋅ ν ) = ℱ K * * ∘ ( ν ⋅ ω ∣ K ( k ) ) = ω ∣ K ( k ) = κ K ∘ ω ∣ K ( k ) .

That is, K is a ω ∣ K -biprojective. Conversely, let ϑ : K ⟶ ( K ⊗ K ˆ ) be a bounded ω ∣ K - K -bimodule map with ℱ K ∘ ϑ = ω ∣ K . Let ( e α ) be a bounded approximate identity for K and we put ν = lim α ϑ ( e α ) . Then, we have

ω ∣ K ( k ) ⋅ ν = lim α ω ∣ K ( k ) ⋅ ϑ ( e α ) = lim α ϑ ( k e α ) = lim α ϑ ( e α k ) = lim α ϑ ( e α ) ⋅ ω ∣ K ( k ) = ν ⋅ ω ( k ) .

And so,

ℱ K ( ν ) ⋅ ω ∣ K ( k ) = lim α ℱ K ∘ ϑ ( e α ) ⋅ ω ∣ K ( k ) = lim α ω ( e α ) ⋅ ω ∣ K ( k ) = ω ∣ K ( k ) .

Therefore, ν is a ω ∣ K -diagonal for K . By Theorem 2.6 [24], K is ω ∣ K -contractible.□

Recall that a linear subspace S 1 ( G ) of the convolution group algebra L 1 ( G ) is said to be a Segal algebra on G if it satisfies the following conditions:

S 1 ( G ) is dense in L 1 ( G ) .
S 1 ( G ) is a Banach space under some norm ‖ . ‖ S and for each f ∈ S 1 ( G )
‖ f ‖ 1 ≤ ‖ f ‖ S .
S 1 ( G ) is left translation invariant and the map x ↦ δ x ∗ f from G into S 1 ( G ) is continuous.
‖ δ x ∗ f ‖ S = ‖ f ‖ S for all f ∈ S 1 ( G ) and x ∈ G .

Every Segal algebra is an abstract Segal algebra with respect to L 1 ( G ) , but the converse is not true; see [17].

Corollary 3

Let G be a locally compact SIN group and S ( G ) be a Segal algebra. If ω ∈ End ( L 1 ( G ) ) and ω ( S ( G ) ) = S ( G ) , then S ( G ) is ω ∣ S ( G ) -biprojective if and only if S ( G ) is ω ∣ S ( G ) -contractible.

Proof

The Segal algebra S ( G ) having an approximate identity follows from [25]. Hence, the proof is complete.□

Theorem 7

Let K be an abstract Segal algebra with respect to ℳ . Suppose K contains a net ( e α ) α such that ( e α ) 2 is an approximate identity for K , ω ( e α ) commutes for any m ∈ ℳ and ω ( K ) ⊆ K . Then, K is ω ∣ K -approximate biprojective if ℳ is ω -biprojective.

Proof

If ℳ is ω -biprojective, then there is a ω - ℳ -bimodule homomorphism ϑ : ℳ ⟶ ( ℳ ⊗ ℳ ˆ ) with ℱ ℳ ∘ ϑ ( m ) = ω ( m ) . We define ϑ α : K ⟶ ( K ⊗ K ˆ ) by ϑ α ( k ) = ϑ ( e α k e α ) , k ∈ K . Clearly, ϑ α is ω ∣ K - K -bimodule homomorphism and we have

□ ℱ K ∘ ϑ α ( k ) = ℱ K ( ϑ ( e α k e α ) ) = ω ( e α k e α ) = ω ( k e α 2 ) → ω ( k ) .

Corollary 4

Let G be a locally compact SIN group and S ( G ) be a Segal algebra. If ω ∈ Hom ( L 1 ( G ) ) and ω ( S ( G ) ) ⊆ S ( G ) and L 1 ( G ) is ω -biprojective, then S ( G ) is ω ∣ S ( G ) -approximate biprojective.

Theorem 8

Let K be an abstract Segal algebra with respect to ℳ . Suppose that K contains a net ( e α ) α such that ( e α ) 2 is an approximate identity for ℳ and m e α = e α m , ( m ∈ ℳ ) . If ω ( K ) ⊆ K and K is ω ∣ K -biprojective, then ℳ is ω -approximate biprojective.

Proof

Suppose that K is ω ∣ K -biprojective. Then, there exists a ω - K -bimodule homomorphism ϑ : K ⟶ ( K ⊗ K ˆ ) with ℱ K ∘ ϑ ( k ) = ω ( k ) ( k ∈ K ) . Take ϑ α ( m ) = ( ι ⊗ ι ) ( ϑ ( e α m e α ) ) , where ι : K → ℳ is the inclusion map. Then, for every m ∈ ℳ , we have

□ ℱ ℳ ∘ ϑ α ( m ) = ℱ ℳ ( ( ι ⊗ ι ) ( ϑ ( e α m e α ) ) ) = ι ( ℱ K ( ϑ ( e α m e α ) ) ) = ι ( ω ( m e α 2 ) ) → ω ( m ) .

3 Conclusion

In this study, we investigated ω ℒ -biprojectivity and ω ¯ -contractibility of Banach algebras. Additionally, we established a relationship between the two definitions. An example is provided to illustrate that the class of ω ℒ -biprojective Banach algebras is larger than that of biprojective Banach algebras. Finally, we examined the concepts of ω -pseudo amenability and ω -approximate biprojectivity in Banach algebras. It is shown that for every unital Banach algebra ℳ , ω -approximate biprojectivity and ω -pseudo contractibility coincide. To further enhance our results, we are interested in exploring the notion of approximate ω ℒ -biprojectivity in Banach algebras in the future.

Acknowledgement

I am grateful to thank the referees for carefully reading the article and for their suggestions that greatly improved its presentation.

Funding information: The author states no funding involved.
Author contributions: The author confirms the sole responsibility for the conception of the study, presented results, and manuscript preparation.
Conflict of interest: The author states no conflict of interest.
Ethical approval: The conducted research is not related to either human or animal use.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2024-09-04

Revised: 2025-02-24

Accepted: 2025-03-07

Published Online: 2025-06-12

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https://doi.org/10.1515/dema-2025-0118

Keywords for this article

Banach algebra; ωℒ-biprojective; ω-contractible

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