Abstract:In this paper, we investigate the notion of approximate biprojectivity for semigroup algebras and for some Banach algebras related to semigroup algebras. We show that ℓ 1 (S) is approximately biprojective if and only if ℓ 1 (S) is biprojective, provided that S is a uniformly locally finite inverse semigroup. Also for a Clifford semigroup S, we show that approximate biprojectivity ℓ 1 (S) * * gives pseudo amenability of ℓ 1 (S). We give a class of Banach algebras related to semigroup algebras which is not appro… Show more
“…Tus, ψ φ ∈ Δ(UP(I, P, A)). Since A is unital by Lemma 2, UP(I, P, A) has a right approximate identity and by [15] Teorem 3.9, we can conclude that UP(I, P, A) is approximately right ψ φ -amenable. Defne a closed ideal in UP(I, P, A) by…”
Section: □ Proposition 1 Let a Be A Weak Approximately Biprojective B...mentioning
In the present paper, we study the approximate biprojectivity and weak approximate biprojectivity of
ℓ
1
-Munn Banach algebras when the related sandwich matrix is regular over
Inv
A
. In fact, we show that a
ℓ
1
-Munn Banach algebra with the regular sandwich matrix over
Inv
A
is approximately biprojective (weak approximately biprojective) if and only if
A
is approximately biprojective (weak approximately biprojective), respectively. We also study approximate biprojectivity of upper triangular Banach algebra when the associated sandwich matrix with elements in
Inv
A
is invertible. Finally, we apply our results to Rees semigroup algebras.
“…Tus, ψ φ ∈ Δ(UP(I, P, A)). Since A is unital by Lemma 2, UP(I, P, A) has a right approximate identity and by [15] Teorem 3.9, we can conclude that UP(I, P, A) is approximately right ψ φ -amenable. Defne a closed ideal in UP(I, P, A) by…”
Section: □ Proposition 1 Let a Be A Weak Approximately Biprojective B...mentioning
In the present paper, we study the approximate biprojectivity and weak approximate biprojectivity of
ℓ
1
-Munn Banach algebras when the related sandwich matrix is regular over
Inv
A
. In fact, we show that a
ℓ
1
-Munn Banach algebra with the regular sandwich matrix over
Inv
A
is approximately biprojective (weak approximately biprojective) if and only if
A
is approximately biprojective (weak approximately biprojective), respectively. We also study approximate biprojectivity of upper triangular Banach algebra when the associated sandwich matrix with elements in
Inv
A
is invertible. Finally, we apply our results to Rees semigroup algebras.
“…G is compact and also we show that L 1 (G, w) is approximately biprojective if and only if G is compact, provided that w ≥ 1 is a continuous weight function, see [21] and [23].…”
Section: Introductionmentioning
confidence: 69%
“…Define ψ ∈ ∆(U P (I, A)) by ψ(a) = φ(a in,in ) for every a = (a i,j ) ∈ U P (I, A). By [23,Theorem 3.9] approximate biprojectivity of U P (I, A) implies that U P (I, A) is left ψ-contractible, then the rest is similar to the proof of Theorem 2.1.…”
Section: A Class Of Matrix Algebra and Approximate Biprojectivitymentioning
We investigate the notions of amenability and its related homological notions for a class of I × I-upper triangular matrix algebra, say U P (I, A), where A is a Banach algebra equipped with a nonzero character. We show that U P (I, A) is pseudo-contractible (amenable) if and only if I is singleton and A is pseudo-contractible (amenable), respectively. We also study the notions of pseudo-amenability and approximate biprojectivity of U P (I, A).Recently approximate versions of amenability and homological properties of Banach algebras have been under more observations. In [24] Zhang introduced the notion of approximately biprojective Banach algebras, that is, A is approximately biprojective if there exists a net of A-bimodule morphism ρAuthor with A. Pourabbas investigated approximate biprojectivity of 2 × 2 upper triangular Banach algebra which is a matrix algebra, also we characterized approximate biprojectivity of Segal algebras and weighted group algebras. We show that a Segal algebra S(G) is approximately biprojective if and only if
“…Indeed, a Banach algebra A is called approximately biprojective if there exists a net (ρ α ) of continuous A-bimodule morphism from A into A ∧ ⊗ A such that π A °ρα (a) ⟶ a for every a ∈ A, where π A : A ∧ ⊗ A ⟶ A is the diagonal operator defined by π A (a ⊗ b) � ab. For recent works about this concept, refer to [8]. roughout the paper, Δ(A) stands for the set of all nonzero multiplicative linear functionals on A. Kaniuth et al [9] introduced the notion of left φ-amenable Banach algebras (φ ∈ Δ(A)) as a generalization of the notion of amenable Banach algebras introduced by Johnson in [10].…”
In this paper, we study the notion of approximate biprojectivity and left
φ
-biprojectivity of some Banach algebras, where
φ
is a character. Indeed, we show that approximate biprojectivity of the hypergroup algebra
L
1
K
implies that
K
is compact. Moreover, we investigate left
φ
-biprojectivity of certain hypergroup algebras, namely, abstract Segal algebras. As a main result, we conclude that (with some mild conditions) the abstract Segal algebra
B
is left
φ
-biprojective if and only if
K
is compact, where
K
is a hypergroup. We also study the approximate biflatness and left
φ
-biflatness of hypergroup algebras in terms of amenability of their related hypergroups.
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