Triangular Banach Algebras Research Articles

Module derivations into iterated duals of triangular Banach algebras

Module derivations into iterated duals of triangular Banach algebras

On σ-amenability of TσA,σB

Let ?A and ?B be two homomorphisms on Banach algebras A and B, respectively. In this paper, we study ?-amenability, ?-weak amenability, ?-biflatness, and ?-biprojectivity of triangular Banach algebras of the form T?A,?B, where ? = ?A ? ?B.

Relative amenability of Banach algebras

Let A be a Banach algebra and I be a closed ideal of A. We say that A is amenable relative to I, if A/I is an amenable Banach algebra. We study the relative amenability of Banach algebras and investigate the relative amenability of triangular Banach algebras and Banach algebras associated to locally compact groups. We generalize some of the previous known results by applying the concept of relative amenability of Banach algebras, especially, we present a generalization of Johnson?s theorem in the concept of relative amenability.

Approximate Biprojectivity of ℓ1‐Munn Banach Algebras

In the present paper, we study the approximate biprojectivity and weak approximate biprojectivity of ℓ1‐Munn Banach algebras when the related sandwich matrix is regular over Inv(A). In fact, we show that a ℓ1‐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.

(2n+1)-weak module amenability of triangular Banach algebras on Inverse semigroup algebras

Let be a commutative (not necessary unital) inverse semigroup with the set of idempotents E ‎then l^1 (S) is a commutative Banach‎ l^1 (E)-module with canonical actions‎. ‎Recently‎, ‎it is shown that the triangular Banach algebra T=Tri(l^1(S), M, l^1(S)) is (n) -weakly l^1 (E)-module amenable‎, ‎provided that M=l^1 (S) and is unital or E satisfies condition D_k for some k∈N ‎. ‎In this paper‎, ‎we show that T is (2n+1)-weakly l^1 (E)-module amenable‎, ‎without any additional conditions on S and E, ‎if M is a certain quotient space of l^1 (S).

CHARACTERIZATION OF SOME BIDERIVATIONS ON TRIANGULAR BANACH ALGEBRAS

Let $A$ and $B$ be unital Banach algebras‎, ‎$X$ be an unital $A-B-$module and $T$ be the triangular Banach algebra associated to $A‎, ‎B$ and $X$‎. The structure of some derivations on triangular Banach algebras was studied by some authors. ‏‎Note that despite the apparent similarity between derivations and biderivations and also inner derivations and inner biderivations‎, ‎there are fundamental differences between them‎. Although there are some studying of biderivations on triangular Banach algebras, any of them do not completely determine the structure of biderivations on triangular Banach algebras. In this paper, we ‎completely characterize biderivations and inner biderivations from $T\times T$ to $T^*$‎ and we show that the first bicohomology group $BH^1(T, T^*)$ is equal to $BH^1(A, A^*)\oplus BH^1(B, B^*)$‎‏.

Approximate biflatness and Johnson pseudo-contractibility of some Banach algebras

We study the structure of Lipschitz algebras under the notions of approximate biflatness and Johnson pseudo-contractibility. We show that for a compact metric space $X$, the Lipschitz algebras ${\rm Lip}_{\alpha}(X)$ and ${\rm lip}_{\alpha}(X)$ are approximately biflat if and only if $X$ is finite, provided that $0<\alpha<1$. We give a necessary and sufficient condition that a vector-valued Lipschitz algebras is Johnson pseudo-contractible. We also show that some triangular Banach algebras are not approximately biflat.

Approximate Connes-biprojectivity of dual Banach algebras

In this paper, we introduce a notion of approximate Connes-biprojectivity for dual Banach algebras. We study the relation between approximate Connes-biprojectivity, approximate Connes amenability and [Formula: see text]-Connes amenability. We propose a criterion to show that certain dual triangular Banach algebras are not approximately Connes-biprojective. Next, we show that for a locally compact group [Formula: see text], the Banach algebra [Formula: see text] is approximately Connes-biprojective if and only if [Formula: see text] is amenable. Finally, for an infinite commutative compact group [Formula: see text], we show that the Banach algebra [Formula: see text] with convolution product is approximately Connes-biprojective, but it is not Connes-biprojective.

Ideal Module Amenability of Triangular Banach Algebras

Let A and B be unital Banach algebras and M be an unital Banach A,B-module. In this paper we define the concept of the (n)-ideal module amenability of Banach algebras and investigate the relation between the (2n-1)-ideal module amenability of triangular Banach algebra Τ = [A M B ] (as a Τ = {[α α] : α ∈u}-module) and (2n - 1)-ideal module amenability of A and B (as an u-module), where u is a (not necessarily unital) Banach algebra such that A, B and M are commutative Banach u-bimodules. Finally, in the case that A = B = M = l 1 (S) and u = l 1 (E), for unital and commutative inverse semigroup S with idempotent set E, we show that T as an u-module is (2n - 1)- ideal module amenable while is not module amenable.

Some Notes on Semidirect Products of Banach Algebras

Let $${\mathcal {A}}$$ and $${\mathcal {U}}$$ be Banach algebras and $${\mathcal {A}} < imes {\mathcal {U}}$$ be the semidirect product of Banach algebras $${\mathcal {A}}$$ and $${\mathcal {U}}$$ where $${\mathcal {A}} < imes {\mathcal {U}}={\mathcal {A}}\oplus {\mathcal {U}}$$ as direct sum of Banach spaces and $${\mathcal {A}}$$ is closed subalgebra and $${\mathcal {U}}$$ is a closed ideal of $${\mathcal {A}} < imes {\mathcal {U}}$$ . The semidirect product $${\mathcal {A}} < imes {\mathcal {U}}$$ is a block of Banach algebras that covers the important classes of Banach algebras such as Lau product Banach algebras and triangular Banach algebras. In this paper we consider this question that under what conditions the semidirect product Banach algebra $${\mathcal {A}} < imes {\mathcal {U}}$$ is a Lau product Banach algebra or a triangular Banach algebra? We find these conditions and answer this question which form the main results of our paper.

Approximate biflatness and approximate biprojectivity of some Banach algebras

In this paper, we study approximate biflatness and approximate biprojectivity of some Banach algebras. In fact, we characterize approximate biflatness of triangular Banach algebras. We also show under which conditions triangular Banach algebras are not approximately biprojective. As an application, we investigate approximate biflatness and approximate biprojectivity of some Banach algebras associated with a locally compact group.

The first module (σ,τ)-cohomology group of triangular Banach algebras of order three

The notion of module amenability for a class of Banach algebras, which could be considered as a generalization of Johnson’s amenability, was introduced by Amini in [Module amenability for semigroup algebras, Semigroup Forum 69 (2004) 243–254]. The weak module amenability of the triangular Banach algebra [Formula: see text], where [Formula: see text] and [Formula: see text] are Banach algebras (with [Formula: see text]-module structure) and [Formula: see text] is a Banach [Formula: see text]-module, is studied by Pourabbas and Nasrabadi in [Weak module amenability of triangular Banach algebras, Math. Slovaca 61(6) (2011) 949–958], and they showed that the weak module amenability of [Formula: see text] triangular Banach algebra [Formula: see text] (as an [Formula: see text]-bimodule) is equivalent with the weak module amenability of the corner algebras [Formula: see text] and [Formula: see text] (as Banach [Formula: see text]-bimodules). The main aim of this paper is to investigate the module [Formula: see text]-amenability and weak module [Formula: see text]-amenability of the triangular Banach algebra [Formula: see text] of order three, where [Formula: see text] and [Formula: see text] are [Formula: see text]-module morphisms on [Formula: see text]. Also, we give some results for semigroup algebras.

Amalgamated duplication of the Banach algebra 𝔄 along a 𝔄-bimodule 𝒜

Let [Formula: see text] and [Formula: see text] be Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with compatible actions. We define the product [Formula: see text], which is a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. After characterization of the multiplier algebra, topological center, (maximal) ideals and spectrum of [Formula: see text], we restrict our investigation to the study of semisimplicity, regularity, Arens regularity of [Formula: see text] in relation to that of the algebras [Formula: see text], [Formula: see text] and the action of [Formula: see text] on [Formula: see text]. We also compute the first cohomology group [Formula: see text] for all [Formula: see text] as well as the first-order cyclic cohomology group [Formula: see text], where [Formula: see text] is the [Formula: see text]th dual space of [Formula: see text] when [Formula: see text] and [Formula: see text] itself when [Formula: see text]. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for Lau products and module extensions of Banach algebras as well as triangular Banach algebra. Finally, special attention is devoted to the cyclic and [Formula: see text]-weak amenability of [Formula: see text]. In this context, several open questions arise.

N-Weak Module Amenability of Triangular Banach Algebras

Abstract Let A, B be Banach A-modules with compatible actions and M be a left Banach A- A-module and a right Banach B- A-module. In the current paper, we study module amenability, n-weak module amenability and module Arens regularity of the triangular Banach algebra - . We employ these results to prove that for an inverse semigroup S with subsemigroup E of idempotents, the triangular Banach algebra is permanently weakly module amenable (as an . As an example, we show that T0 is T0-module Arens regular if and only if the maximal group homomorphic image GS of S is finite.

(2n − 1)-Ideal amenability of triangular banach algebras

Let \(\mathcal {A}\) and \(\mathcal {B}\) be two unital Banach algebras and let \(\mathcal {M}\) be an unital Banach \(\mathcal {A}, \mathcal {B}\)-module. Also, let \(\mathcal {T}=\left [\begin {array}{cc} \mathcal {A} & \mathcal {M} \\ & \mathcal {B} \end {array}\right ]\) be the corresponding triangular Banach algebra. Forrest and Marcoux (Trans. Amer. Math. Soc. 354 (2002) 1435–1452) have studied the n-weak amenability of triangular Banach algebras. In this paper, we investigate (2n−1)-ideal amenability of \(\mathcal {T}\) for all n≥1. We introduce the structure of ideals of these Banach algebras and then, we show that (2n−1)-ideal amenability of \(\mathcal {T}\) depends on (2n−1)-ideal amenability of Banach algebras \(\mathcal {A}\) and \(\mathcal {B}\).

Weak module amenability of triangular Banach algebras

Abstract Let A and B be unital Banach algebras and let M be a unital Banach A,B-module. Forrest and Marcoux [6] have studied the weak amenability of triangular Banach algebra $\mathcal{T} = \left[ {_B^{AM} } \right]$ and showed that T is weakly amenable if and only if the corner algebras A and B are weakly amenable. When $\mathfrak{A}$ is a Banach algebra and A and B are Banach $\mathfrak{A}$-module with compatible actions, and M is a commutative left Banach $\mathfrak{A}$-A-module and right Banach $\mathfrak{A}$-B-module, we show that A and B are weakly $\mathfrak{A}$-module amenable if and only if triangular Banach algebra T is weakly $\mathfrak{T}$-module amenable, where $\mathfrak{T}: = \{ [^\alpha _\alpha ]:\alpha \in \mathfrak{A}\} $.

Jordan [formula omitted]-derivations on triangular algebras and related mappings

Let R be a 2-torsion free commutative ring with identity, A,B be unital algebras over R and M be a unital (A,B)-bimodule, which is faithful as a left A-module and also as a right B-module. Let T=AM0B be the triangular algebra consisting of A,B and M, and let d be an R-linear mapping from T into itself. Suppose that A and B have only trivial idempotents. Then the following statements are equivalent: (1) d is a Jordan (α,β)-derivation on T; (2) d is a Jordan triple (α,β)-derivation on T; (3) d is an (α,β)-derivation on T. Furthermore, a generalized version of this result is also given. We characterize the actions of automorphisms and skew derivations on the triangular algebra T. The structure of continuous (α,β)-derivations of triangular Banach algebras and that of generalized Jordan (α,β)-derivations of upper triangular matrix algebras are described.

A SIMPLE PROOF OF A THEOREM ON (2n)-WEAK AMENABILITY

A simple proof of (2n)-weak amenability of the triangular Banach algebra T= [(A A) (0 A)] is given where A is a unital C*-algebra.

Weak amenability of triangular Banach algebras

Let A \mathcal {A} and B \mathcal {B} be unital Banach algebras, and let M \mathcal {M} be a Banach A , B \mathcal {A},\mathcal {B} -module. Then T = [ A a m p ; M 0 a m p ; B ] \mathcal {T} = \begin {bmatrix}\mathcal {A} &amp; \mathcal {M}\\ 0 &amp; \mathcal {B} \end {bmatrix} becomes a triangular Banach algebra when equipped with the Banach space norm ‖ [ a a m p ; m 0 a m p ; b ] ‖ = ‖ a ‖ A + ‖ m ‖ M + ‖ b ‖ B \left \Vert \begin {bmatrix} a &amp; m\\ 0 &amp; b \end {bmatrix} \right \Vert = \Vert a \Vert _{\mathcal {A}} + \Vert m \Vert _{\mathcal {M}} + \Vert b \Vert _{\mathcal {B}} . A Banach algebra T \mathcal {T} is said to be n n -weakly amenable if all derivations from T \mathcal {T} into its n t h n^{\mathrm {th}} dual space T ( n ) \mathcal {T}^{(n)} are inner. In this paper we investigate Arens regularity and n n -weak amenability of a triangular Banach algebra T \mathcal {T} in relation to that of the algebras A \mathcal {A} , B \mathcal {B} and their action on the module M \mathcal {M} .

Isometric isomorphisms of triangular Banach algebras

Let and ℬ be Banach algebras. Let ℳ be a Banach with bounded 1. Then is a Banach algebra with the usual operations and the norm $$ \left\| {\left[ {_0^A } \right.} \right.\left. {\left. {_B^M } \right]} \right\| = \left\| A \right\| + \left\| M \right\| + \left\| B \right\| $$ . Such an algebra is called a triangular Banach algebra. In this paper the isometric isomorphisms of triangular Banach algebras are characterized.