• Honam Mathematical Journal
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• v.42 no.3
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• pp.537-550
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• 2020

Let A, B, 𝔘 be Banach algebras and B be a Banach 𝔘-bimodule also A be a Banach B-𝔘-module. In this paper we study the relation between module amenability, weak module amenability and module approximate amenability of module Lau product A × _α B and that of Banach algebras A, B.

• Honam Mathematical Journal
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• v.43 no.2
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• pp.209-219
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• 2021

We define a new concept of module amenability which is compatible with original definition of amenability. For a module dual algebra 𝓐, we will show that if every module derivation D : 𝓐** → J𝓐**⊥ is inner then 𝓐 is weak module amenable. Moreover, we will prove that under certain conditions, weak module amenability of 𝓐** implies weak module amenability of 𝓐.

• Honam Mathematical Journal
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• v.42 no.1
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• pp.37-48
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• 2020

We first define a new multiplication on triangular Banach algebras. Then we study amenability, weak amenability, biflatness and biprojectivity of these algebras.

• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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• pp.1109-1124
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• 2018

For a metric space (X, d) and ${\alpha}$ > 0. We study the structure and properties of vector-valued Lipschitz algebra $Lip{\alpha}(X,E)$ and $lip{\alpha}(X,E)$ of order ${\alpha}$. We investigate the approximate and Character amenability of vector-valued Lipschitz algebras.

• Bulletin of the Korean Mathematical Society
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• v.52 no.3
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• pp.761-769
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• 2015

In this study, the character amenability of Banach algebras is considered and some characterization theorems are established. Indeed, we prove that the character amenability of Lipschitz algebras is equivalent to that of Banach algebras.

• Honam Mathematical Journal
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• v.46 no.1
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• pp.1-11
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• 2024

In this paper, we introduce the notion of 𝜎-Jordan amenability of Banach algebras and some hereditary are investigated. Similar to Johnson's classic result, we give the notions of 𝜎-Jordan approximate and 𝜎-Jordan virtual diagonals, and find some relations between the existence of them and 𝜎-Jordan amenability.

• The Pure and Applied Mathematics
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• v.31 no.1
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• pp.1-19
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• 2024

In this paper, the notions of strong Connes amenability for certain products of Banach algebras and module extension of dual Banach algebras is investigated. We characterize χ ⊗ η-strong Connes amenability of projective tensor product ${\mathbb{K}}{\hat{\bigotimes}}{\mathbb{H}}$ via χ ⊗ η-σwc virtual diagonals, where χ ∈ 𝕂_* and η ∈ ℍ_* are linear functionals on dual Banach algebras 𝕂 and ℍ, respectively. Also, we present some conditions for the existence of (χ, θ)-σwc virtual diagonals in the θ-Lau product of 𝕂 ×_θ ℍ. Finally, we characterize the notion of (χ, 0)-strong Connes amenability for module extension of dual Banach algebras 𝕂 ⊕ 𝕏, where 𝕏 is a normal Banach 𝕂-bimodule.

• Bulletin of the Korean Mathematical Society
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• v.47 no.2
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• pp.307-317
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• 2010

In this paper we find necessary and sufficient conditions for weak amenability of the convolution Banach algebras A(K) and $L^2(K)$ for a compact hypergroup K, together with their applications to convolution Banach algebras $L^p(K)$ ($2\;{\leq}\;p\;<\;{\infty}$). It will further be shown that the convolution Banach algebra A(G) for a compact group G is weakly amenable if and only if G has a closed abelian subgroup of finite index.

• Communications of the Korean Mathematical Society
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• v.35 no.3
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• pp.891-906
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• 2020

In this paper, the commutative module amenable Banach algebras are characterized. The hereditary and permanence properties of module amenability and the relations between module amenability of a Banach algebra and its ideals are explored. Analogous to the classical case of amenability, it is shown that the projective tensor product and direct sum of module amenable Banach algebras are again module amenable. By an application of Ryll-Nardzewski fixed point theorem, it is shown that for an inverse semigroup S, every module derivation of 𝑙¹(S) into a reflexive module is inner.

• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.1991-1999
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• 2017

Let A and B be two Banach algebras and $T:B{\rightarrow}A$ be a bounded homomorphism, with ${\parallel}T{\parallel}{\leq}1$. Recently, Dabhi, Jabbari and Haghnejad Azar (Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 9, 1461-1474) obtained some results about the n-weak amenability of $A{\times}_TB$. In the present paper, we address a gap in the proof of these results and extend and improve them by discussing general necessary and sufficient conditions for $A{\times}_TB$ to be n-weakly amenable, for an integer $n{\geq}0$.