• 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.1
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• pp.68-77
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• 2021

This paper considers pseudo Jordan homomorphisms on module extensions of Banach algebras and triangular Banach algebras. We characterize pseudo Jordan homomorphisms on module extensions of Banach algebras and triangular Banach algebras. Moreover, we define pseudo derivations on the above stated Banach algebras and characterize this new notion on those algebras.

• 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.40 no.1
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• pp.53-61
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• 2003

We prove the generalized Hyers-Ulam-Rassias stability of the invertible mapping in a Banach module over a unital Banach algebra, and prove the generalized Hyers-Ulam-Rassias stability of the Jensen's functional equations in a Hilbert module over a unital $C^{*}$-algebra.

• Honam Mathematical Journal
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• v.41 no.2
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• pp.357-368
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• 2019

We define the concepts of the first and the second module dual of a Banach space X. And also bring a new concept of module amenability for a Banach algebra ${\mathcal{A}}$. For inverse semigroup S, we will give a new action for ${\ell}^1(S)$ as a Banach ${\ell}^1(E_S)$-module and show that if S is amenable then ${\ell}^1(S)$ is ${\ell}^1(E_S)$-module amenable.

• Bulletin of the Korean Mathematical Society
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• v.47 no.4
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• pp.663-673
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• 2010

This work was intended as an attempt to introduce and investigate the Connes-amenability of module extension of dual Banach algebras. It is natural to try to study the $weak^*$-continuous derivations on the module extension of dual Banach algebras and also the weak Connes-amenability of such Banach algebras.

• 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.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.2
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• pp.221-238
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• 2011

We prove the Hyers-Ulam stability of generalized Jensen's equations in Banach modules over a unital $C^{\ast}$-algebra. It is applied to show the stability of generalized Jensen's equations in a Hilbert module over a unital $C^{\ast}$-algebra. Moreover, we prove the stability of linear operators in a Hilbert module over a unital $C^{\ast}$-algebra.

• Journal of the Chungcheong Mathematical Society
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• v.19 no.1
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• pp.9-18
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• 2006

In this paper, the completion of a quasi-normed algebra and the completion of a quasi-normed module over a Banach algebra are constructed.

• Journal of the Chungcheong Mathematical Society
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• v.14 no.1
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• pp.113-122
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• 2001

We prove the Hyers-Ulam-Rassias stability of the Jensen's equation in Banach modules over a Banach algebra.