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Journal articles on the topic 'Banach module'

Author: Grafiati

Published: 2 June 2025

Last updated: 31 July 2025

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1

Ebrahimi Bagha, D., and M. Amini. "Module amenability for Banach modules." Cubo (Temuco) 13, no. 2 (2011): 127–37. http://dx.doi.org/10.4067/s0719-06462011000200007.

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2

Soltanmoradi, Shabani, Davood Ebrahimi Bagha, and Pourbahri Rahpeyma. "Weak module amenability for the second dual of a Banach algebra." Acta et Commentationes Universitatis Tartuensis de Mathematica 25, no. 2 (2021): 297–306. http://dx.doi.org/10.12697/acutm.2021.25.19.

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In this paper we study the weak module amenability of Banach algebras which are Banach modules over another Banach algebra with compatible actions. We show that for every module derivation D : A ↦ ( A/J_A )∗ if D∗∗(A∗∗) ⊆ WAP (A/J_A ), then weak module amenability of A∗∗ implies that of A. Also we prove that under certain conditions for the module derivation D, if A∗∗ is weak module amenable then A is also weak module amenable.

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3

Bami, Mahmood Lashkarizadeh, Mohammad Valaei та Massoud Amini. "The Structure ofφ-Module Amenable Banach Algebras". Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/176736.

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We study the concept ofφ-module amenability of Banach algebras, which are Banach modules over another Banach algebra with compatible actions. Also, we compare the notions ofφ-amenability andφ-module amenability of Banach algebras. As a consequence, we show that, ifSis an inverse semigroup with finite setEof idempotents andl1Sis a commutative Banachl1E-module, thenl1S**isφ**-module amenable if and only ifSis finite, whenφ∈Homl1El1Sis an epimorphism. Indeed, we have generalized a well-known result due to Ghahramani et al. (1996).

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4

Tavkoli, Shirin, Rasoul Abazari та Ali Jabbari. "Simple Proofs for Bochner-Schoenberg-Eberlein and the Bochner-Schoenberg-Eberlein Module Properties on ℓpX,A". Journal of Function Spaces 2024 (2 травня 2024): 1–10. http://dx.doi.org/10.1155/2024/5893357.

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Let X be a nonempty set, A be a commutative Banach algebra, and 1≤p<∞. In this paper, we present a concise proof for the result concerning the BSE (Banach space extension) property of ℓpX,A. Specifically, we establish that ℓpX,A possesses the BSE property if and only if X is finite and A is BSE. Additionally, we investigate the BSE module property on Banach ℓpX,A-modules and demonstrate that a Banach space ℓpX,Y serves as a BSE Banach ℓpX,A-module if and only if X is finite and Y represents a BSE Banach A-module.

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5

Pugach, L. I. "On the C-Projectivity of ideals in Banach algebras." Glasgow Mathematical Journal 40, no. 2 (1998): 143–45. http://dx.doi.org/10.1017/s0017089500032456.

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The notion of projective Banach module was defined by Helemskii in [1]—the paper which properly founded the homological theory of Banach algebras. The same author introduced the definition of the (relatively) flat Banach module in [2]. Recently M. C. White [3] modified both of those definitions, introducing so called C-projective and C-flat Banach modules.

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6

Hill, Terje, and David A. Robbins. "Module bundles and module amenability." Acta et Commentationes Universitatis Tartuensis de Mathematica 25, no. 1 (2021): 119–41. http://dx.doi.org/10.12697/acutm.2021.25.08.

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Let X be a compact Hausdorff space, and let {Ax : x ∈ X} and {Bx : x ∈ X} be collections of Banach algebras such that each Ax is a Bx-bimodule. Using the theory of bundles of Banach spaces as a tool, we investigate the module amenability of certain algebras of Ax-valued functions on X over algebras of Bx-valued functions on X.

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7

Ghorbani, M., and D. Ebrahimi Bagha. "Weak Module Amenability of Module Extension Banach Algebras." International Journal of Contemporary Mathematical Sciences 20, no. 1 (2025): 19–28. https://doi.org/10.12988/ijcms.2025.91978.

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In this paper we study module and weak module amenability of the module extension Banach algebra $A\oplus X$ of a Banach algebra A by a Banach A-module X. As an example we show that for an inverse semigroup S with set of idempotents E, the module extension ${\ell ^{1}}(E)\oplus {\ell ^{1}}(S)$ is amenable as an ${\ell ^{1}}(E)$-module iff S is amenable. We also study module biflatness and module biprojectivity of module extensions.

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8

Journal, Baghdad Science. "On Fully Stable Banach Algebra Modules and Fully Pesudo Stable Banach Algebra Modules." Baghdad Science Journal 15, no. 1 (2018): 102–5. http://dx.doi.org/10.21123/bsj.15.1.102-105.

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The concept of fully pseudo stable Banach Algebra-module (Banach A-module) which is the generalization of fully stable Banach A-module has been introduced. In this paper we study some properties of fully stable Banach A-module and another characterization of fully pseudo stable Banach A-module has been given.

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9

Alikahi, Mahdieh, and Mohammad Ramezanpour. "Module derivations into iterated duals of triangular Banach algebras." Filomat 38, no. 1 (2024): 119–27. http://dx.doi.org/10.2298/fil2401119a.

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Let A be a Banach algebra, A and B be Banach A-module with compatible actions and X be a Banach left A-A-module and Banach right B-A-module. Then the corresponding triangular Banach algebra Tri(A,X, B) is a Banach A-module with compatible actions. In this paper, we study n-weak module amenability of module extension Banach algebras to provide necessary and sufficient conditions for n-weak module amenability (as an A-module) of Tri(A,X, B), when A and B are not necessarily unital and not have bounded approximate identity. This not only fixes the gaps in some known results in the literature but

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10

Ghorbai, M., and Davood Ebrahimi Bagha. "Amenability of A⊕_T X as an extension of Banach algebra." Mathematica Montisnigri 49 (2020): 39–48. http://dx.doi.org/10.20948/mathmontis-2020-49-3.

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Let 𝐴𝐴,𝑋𝑋,𝔘𝔘 be Banach algebras and 𝐴𝐴 be a Banach 𝔘𝔘-bimodule also 𝑋𝑋 be a Banach 𝐴𝐴−𝔘𝔘-module. In this paper we study the relation between module amenability, weak module amenability and module approximate amenability of Banach algebra 𝐴𝐴⊕𝑇𝑇𝑋𝑋 and that of Banach algebras 𝐴𝐴,𝑋𝑋. Where 𝑇𝑇: 𝐴𝐴×𝐴𝐴→𝑋𝑋 is a bounded bi-linear mapping with specificconditions.

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11

İnceboz, Hülya, та Berna Arslan. "The first module (σ,τ)-cohomology group of triangular Banach algebras of order three". Journal of Algebra and Its Applications 17, № 12 (2018): 1850225. http://dx.doi.org/10.1142/s0219498818502250.

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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,

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12

Sadeghi, H., and Bami Lashkarizadeh. "Module amenability, module character biprojectivity and module character biflatness of lau product of two Banach algebras." Filomat 32, no. 19 (2018): 6627–41. http://dx.doi.org/10.2298/fil1819627s.

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Let A be a Banach algebra and T be an U-module homomorphism from U-bimodule B into U-bimodule A. We investigate module amenability (resp. module approximate amenability), module character amenability (resp. module character approximate amenability), module character biprojectivity and module character biflatness of A x Tu B for every two Banach U-bimodule A and B.

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13

Nasrabadi, Ebrahim. "Weak module amenability of triangular banach algebras II." Mathematica Slovaca 69, no. 2 (2019): 425–32. http://dx.doi.org/10.1515/ms-2017-0234.

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Abstract Let A and B be Banach 𝔄-bimodule and Banach 𝔅-bimodule algebras, respectively. Also let M be a Banach A, B-module and Banach 𝔄, 𝔅-module with compatible actions. In the case of 𝔄 = 𝔅, the author along with Pourabbas [5] have studied the weak 𝔄-module amenability of triangular Banach algebra $\begin{array}{} \displaystyle \mathcal{T}=\left[\begin{array}{rr} A & M \\ & B \end{array} \right] \end{array}$ and showed that 𝓣 is weakly 𝔄-module amenable if and only if the corner Banach algebras A and B are weakly 𝔄-module amenable, where A, B and M are unital. In this paper we invest

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14

Frank, Michael, and Alexander A. Pavlov. "Module weak Banach-Saks and module Schur properties of Hilbert C*-modules." Journal of Operator Theory 70, no. 1 (2103): 53–73. http://dx.doi.org/10.7900/jot.2011apr21.1933.

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15

Iwata, Yoritaka. "Theory of B(X)-Module: Algebraic Module Structure of Generally Unbounded Infinitesimal Generators." Advances in Mathematical Physics 2020 (November 29, 2020): 1–27. http://dx.doi.org/10.1155/2020/3989572.

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The concept of logarithmic representation of infinitesimal generators is introduced, and it is applied to clarify the algebraic structure of bounded and unbounded infinitesimal generators. In particular, by means of the logarithmic representation, the bounded components can be extracted from generally unbounded infinitesimal generators. In conclusion, the concept of module over a Banach algebra is proposed as the generalization of the Banach algebra. As an application to mathematical physics, the rigorous formulation of a rotation group, which consists of unbounded operators being written by d

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16

Magajna, Bojan. "The minimal operator module of a Banach module." Proceedings of the Edinburgh Mathematical Society 42, no. 1 (1999): 191–208. http://dx.doi.org/10.1017/s0013091500020113.

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17

Bodaghi, A., A. Teymouri, and D. Ebrahimi Bagha. "Derivations on the module extension Banach algebras." Ukrains’kyi Matematychnyi Zhurnal 73, no. 4 (2021): 566–76. http://dx.doi.org/10.37863/umzh.v73i4.240.

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UDC 517.986 We correct some results presented in [M. Eshaghi Gordji, F. Habibian, A. Rejali, <em> Ideal amenability of module extension Banach algebras</em>, Int. J. Contemp. Math. Sci., <strong>2</strong>, No. 5, 213–219 (2007)] and, using the obtained consequences, we find necessary and sufficient conditions for the module extension to be -weakly amenable, where is a closed ideal of the Banach algebra and is a closed -submodule of the Banach -bimodule We apply this result to the module extension where are two Banach -bimodules.

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18

Ghahramani, F., and J. P. Mcclure. "Module Homomorphisms of the Dual Modules of Convolution Banach Algebras." Canadian Mathematical Bulletin 35, no. 2 (1992): 180–85. http://dx.doi.org/10.4153/cmb-1992-026-8.

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AbstractSuppose that A is either the group algebra L1 (G) of a locally compact group G, or the Volterra algebra or a weighted convolution algebra with a regulated weight. We characterize: a) Module homomorphisms of A*, when A* is regarded an A** left Banach module with the Arens product, b) all the weak*-weak* continuous left multipliers of A**.

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19

Hõim, Terje, and D. A. Robbins. "Sectional representation of Banach modules and their multipliers." International Journal of Mathematics and Mathematical Sciences 2003, no. 13 (2003): 817–25. http://dx.doi.org/10.1155/s0161171203207109.

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LetXbe a Banach module over the commutative Banach algebraAwith maximal ideal spaceΔ. We show that there is a norm-decreasing representation ofXas a space of bounded sections in a Banach bundleπ:ℰ→Δ, whose fibers are quotient modules ofX. There is also a representation ofM(X), the space of multipliersT:A→X, as a space of sections in the same bundle, but this representation may not be continuous. These sectional representations subsume results of various authors over the past three decades.

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20

Ettefagh, Mina. "Biprojectivity and biflatness of generalized module extension Banach algebras." Filomat 32, no. 17 (2018): 5895–905. http://dx.doi.org/10.2298/fil1817895e.

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We investigate biprojectivity and biflatness of generalized module extension Banach algebra A Z B, in which A and B are Banach algebras and B is an algebraic Banach A-bimodule, with multiplication: (a, b)?(a',b') = (aa', ab' + ba' + bb')

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21

Chilin, V. I., and J. A. Karimov. "The Cyclical Compactness in Banach C∞(Q)-Modules." Contemporary Mathematics. Fundamental Directions 65, no. 1 (2019): 137–55. http://dx.doi.org/10.22363/2413-3639-2019-65-1-137-155.

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In this paper, we study the class of laterally complete commutative unital regular algebras A over arbitrary ﬁelds. We introduce a notion of passport Γ(X) for a faithful regular laterally complete A- modules X, which consist of uniquely deﬁned partition of unity in the Boolean algebra of all idempotents in A and of the set of pairwise diﬀerent cardinal numbers. We prove that A-modules X and Y are isomorphic if and only if Γ(X)= Γ(Y ). Further we study Banach A-modules in the case A = C∞(Q) or A = C∞(Q)+ i · C∞(Q). We establish the equivalence of all norms in a ﬁnite-dimensional (respectively,

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22

Fozouni, Mohammad. "Generalized Injectivity of Banach Modules." Sarajevo Journal of Mathematics 11, no. 2 (2024): 197–204. http://dx.doi.org/10.5644/sjm.11.2.06.

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In this paper, we study the notion of $\phi$-injectivity in the special case that $\phi=0$. For an arbitrary locally compact group $G$, we characterize the 0-injectivity of $L^{1}(G)$ as a left $L^{1}(G)$ module. Also, we show that $L^{1}(G)^{**}$ and $L^{p}(G)$ for $1<p<\infty$ are 0-injective Banach $L^{1}(G)$ modules.

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23

Journal, Baghdad Science. "On Fully Stable Banach Algebra Modules Relative to an Ideal." Baghdad Science Journal 14, no. 4 (2017): 813–15. http://dx.doi.org/10.21123/bsj.14.4.813-815.

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In this paper, the concept of fully stable Banach Algebra modules relative to an ideal has been introduced. Let A be an algebra, X is called fully stable Banach A-module relative to ideal K of A, if for every submodule Y of X and for each multiplier ?:Y?X such that ?(Y)?Y+KX. Their properties and other characterizations for this concept have been studied.

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24

Sahleh, Abbas, and Abbas Zivari-Kazempour. "Arens Regularity of Certain Class of Banach Algebras." Abstract and Applied Analysis 2011 (2011): 1–6. http://dx.doi.org/10.1155/2011/680952.

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We study Arens regularity of the left and right module actions of on , where is thenth dual space of a Banach algebra , and then investigate (quotient) Arens regularity of as a module extension of Banach algebras.

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25

Gordji, Madjid Eshaghi, Fereydoun Habibian, and Ali Rejali. "MODULE EXTENSION OF DUAL BANACH ALGEBRAS." Bulletin of the Korean Mathematical Society 47, no. 4 (2010): 663–73. http://dx.doi.org/10.4134/bkms.2010.47.4.663.

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26

Ghaffari, Ali. "Module Homomorphisms Associated with Banach Algebras." Taiwanese Journal of Mathematics 15, no. 3 (2011): 1075–88. http://dx.doi.org/10.11650/twjm/1500406285.

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27

Bodaghi, Abasalt, and Massoud Amini. "Module character amenability of Banach algebras." Archiv der Mathematik 99, no. 4 (2012): 353–65. http://dx.doi.org/10.1007/s00013-012-0430-y.

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28

Ozawa, Masanao. "Boolean valued interpretation of Banach space theory and module structures of von Neumann algebras." Nagoya Mathematical Journal 117 (March 1990): 1–36. http://dx.doi.org/10.1017/s0027763000001793.

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Recently, systematic applications of the Scott-Solovay Boolean valued set theory were done by several authors; Takeuti [25, 26, 27, 28, 29, 30], Nishimura [13, 14] Jech [8] and Ozawa [15, 16, 17, 18, 19, 20] in analysis and Smith [23], Eda [2, 3] in algebra. This approach seems to be providing us with a new and powerful machinery in analysis and algebra. In the present paper, we shall study Banach space objects in the Scott-Solovay Boolean valued universe and provide some useful transfer principles from theorems of Banach spaces to theorems of Banach modules over commutative AW*-algebras. The

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29

Ettefagh, Mina. "Bounded approximate version of module character contractibility of Banach algebras." Filomat 37, no. 23 (2023): 7741–59. http://dx.doi.org/10.2298/fil2323741e.

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The (bounded) approximate version of module character contractibility of Banach algebras is introduced and studied. This new concept is characterized by several different concepts such as bounded approximate module character diagonals. Moreover, this new concept is investigated for second dual, unitization, tensor product and lp-direct sums of Banach algebras.

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30

Ettefagh, Mina. "Bounded version of approximate module character amenability of Banach algebras." Filomat 37, no. 24 (2023): 8079–93. http://dx.doi.org/10.2298/fil2324079e.

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The bounded version of approximate module character amenability of Banach algebras is introduced and studied. This new concept is characterized by several different concepts such as bounded approximate module character means. Moreover, this new concept is investigated for second dual, unitization, tensor product and lp-direct sums of Banach algebras.

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31

Crabb, M. J., J. Duncan, and C. M. McGregor. "On one-sided primitivity of Banach algebras." Proceedings of the Edinburgh Mathematical Society 53, no. 1 (2010): 111–23. http://dx.doi.org/10.1017/s0013091508000783.

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AbstractLet S be the semigroup with identity, generated by x and y, subject to y being invertible and yx = xy2. We study two Banach algebra completions of the semigroup algebra ℂS. Both completions are shown to be left-primitive and have separating families of irreducible infinite-dimensional right modules. As an appendix, we offer an alternative proof that ℂS is left-primitive but not right-primitive. We show further that, in contrast to the completions, every irreducible right module for ℂS is finite dimensional and hence that ℂS has a separating family of such modules.

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32

Ebadian, Ali, and Ali Jabbari. "Biprojectivity and biflatness of amalgamated duplication of Banach algebras." Journal of Algebra and Its Applications 19, no. 07 (2019): 2050132. http://dx.doi.org/10.1142/s0219498820501327.

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Let [Formula: see text] and [Formula: see text] be two Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with the left and right compatible action of [Formula: see text] on [Formula: see text]. Let [Formula: see text] be a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. We show that (super) amenability of [Formula: see text] implies (super) module amenability of [Formula: see text] and (super) amenability [Formula: see text]. We investigate biprojectivity and biflatness of [Formula: see text] in the some especial c

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33

Helemskii, A. Ya. "Extreme Version of Projectivity for Normed Modules Over Sequence Algebras." Canadian Journal of Mathematics 65, no. 3 (2013): 559–74. http://dx.doi.org/10.4153/cjm-2012-006-2.

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AbstractWe define and study the so-called extreme version of the notion of a projective normed module. The relevant definition takes into account the exact value of the norm of the module in question, in contrast with the standard known definition that is formulated in terms of normtopology.After the discussion of the case where our normed algebra A is just C, we concentrate on the case of the next degree of complication, where A is a sequence algebra satisfying some natural conditions. The main results give a full characterization of extremely projective objects within the subcategory of the

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34

Jabbari, Ali, and Ali Ebadian. "Bi-derivations and quasi-multipliers on module extensions Banach algebras." Boletim da Sociedade Paranaense de Matemática 41 (December 26, 2022): 1–10. http://dx.doi.org/10.5269/bspm.52574.

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This paper characterize two bi-linear maps bi-derivations and quasi-multipliers on the module extension Banach algebra $A\oplus_1 X$, where $A$ is a Banach algebra and $X$ is a Banach $A$-module. Under some conditions, it is shown that if every bi-derivation on $A\oplus_1 A$ is inner, then the quotient group of bounded bi-derivations and inner bi-derivations, is equal to space of quasi-multipliers of $A$. Moreover, it is proved that $\mathrm{QM}(A \oplus_1 A)=\mathrm{QM}(A)\oplus (\mathrm{QM}(A)+\mathrm{QM}(A)')$, where $\mathrm{QM}(A)'=\{m\in \mathrm{QM}(A):m(0,a)=m(a,0)=0\}$.

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35

Capitão, Pedro, and Lina Oliveira. "Lie Modules of Banach Space Nest Algebras." Mathematics 12, no. 8 (2024): 1251. http://dx.doi.org/10.3390/math12081251.

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In the present work, we extend to Lie modules of Banach space nest algebras a well-known characterisation of Lie ideals of (Hilbert space) nest algebras. Let A be a Banach space nest algebra and L be a weakly closed Lie A-module. We show that there exist a weakly closed A-bimodule K, a weakly closed subalgebra DK of A, and a largest weakly closed A-bimodule J contained in L,such that J⊆L⊆K+DK, with [K,A]⊆L. The first inclusion holds in general, whilst the second is shown to be valid in a class of nest algebras.

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36

MOHAMMADZADEH, S., and H. R. E. VISHKI. "ARENS REGULARITY OF MODULE ACTIONS AND THE SECOND ADJOINT OF A DERIVATION." Bulletin of the Australian Mathematical Society 77, no. 3 (2008): 465–76. http://dx.doi.org/10.1017/s0004972708000403.

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AbstractIn this paper, we give a simple criterion for the Arens regularity of a bilinear mapping on normed spaces, which applies in particular to Banach module actions, and then investigate those conditions under which the second adjoint of a derivation into a dual Banach module is again a derivation. As a consequence of the main result, a simple and direct proof for several older results is also included.

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37

Ilka, Elham, Amin Mahmoodi, and Abasalt Bodaghi. "Some module cohomological properties of Banach algebras." Mathematica Bohemica 145, no. 2 (2019): 127–40. http://dx.doi.org/10.21136/mb.2019.0055-17.

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38

Teymouri, A., A. Bodaghi, and D. Ebrahimi Bagha. "Derivations on the Module Extension Banach Algebras." Ukrainian Mathematical Journal 73, no. 4 (2021): 661–73. http://dx.doi.org/10.1007/s11253-021-01950-x.

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39

GrønbaeK, Niels. "Commutative Banach Algebras, Module Derivations, and Semigroups." Journal of the London Mathematical Society s2-40, no. 1 (1989): 137–57. http://dx.doi.org/10.1112/jlms/s2-40.1.137.

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40

Gordji, M. Eshaghi, F. Habibian, and A. Rejali. "Ideal amenability of module extension Banach algebras." International Journal of Contemporary Mathematical Sciences 2 (2007): 213–19. http://dx.doi.org/10.12988/ijcms.2007.07014.

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41

Hosseiniun, S. A., and D. Ebrahimi Bagha. "The structure of module amenable Banach algebras." International Mathematical Forum 2 (2007): 237–41. http://dx.doi.org/10.12988/imf.2007.07022.

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42

Sadeghi, H., and M. Lashkarizadeh Bami. "Module character inner amenability of Banach algebras." Mathematical Sciences 11, no. 3 (2017): 173–79. http://dx.doi.org/10.1007/s40096-017-0210-8.

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43

Azaraien, Hojat, and Davood Ebrahimi Bagha. "Ideal amenability of module Lau Banach algebra." Indagationes Mathematicae 29, no. 2 (2018): 738–45. http://dx.doi.org/10.1016/j.indag.2017.12.003.

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44

Gouda, Y. Gh. "On the cohomology of Banach A∞-module over admissible Banach A∞-algebra." Journal of the Egyptian Mathematical Society 20, no. 2 (2012): 53–56. http://dx.doi.org/10.1016/j.joems.2012.08.006.

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45

Hamouda, Hawa Alsanousi. "Module Maps and Invariant Subsets of Banach Modules of Locally Compact Groups." Bulletin of the Belgian Mathematical Society - Simon Stevin 21, no. 2 (2014): 253–61. http://dx.doi.org/10.36045/bbms/1400592623.

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46

Apostol, Constantin, and Lawrence Fialkow. "Structural Properties of Elementary Operators." Canadian Journal of Mathematics 38, no. 6 (1986): 1485–524. http://dx.doi.org/10.4153/cjm-1986-072-6.

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Let and denote complex Banach algebras and let b e a left Banach module and a right Banach -module. Ifwe define the bounded linear elementary operator R(A, B), acting on , byFor the case , elementary operators were introduced by Lumer and Rosenblum [19], who studied their spectral properties. In this setting many authors subsequently studied spectral, algebraic, metric, and structural properties of elementary operators, with particular attention devoted to the inner derivations δa (δa(x) = ax – xa) [25], generalized derivations τ(a, b) (τ(a, b)(x) = ax – xb) [9, 10], and elementary multiplicat

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47

Arslan, Berna. "On generalized biderivations of Banach algebras." AIMS Mathematics 9, no. 12 (2024): 36259–72. https://doi.org/10.3934/math.20241720.

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<p>The aim of this paper is to introduce the concept of generalized biderivations of unital Banach algebras and prove some results concerning generalized biamenability of unital Banach algebras. Let $ A $ and $ B $ be unital Banach algebras, and let $ X $ be a unital $ A $-$ B $-module. Let $ T = Tri(A, X, B) $ be the corresponding triangular Banach algebra. We also study the generalized biamenability of triangular Banach algebras and show that if $ X = \{0\} $ and $ T $ is generalized biamenable, then $ A $ and $ B $ are both generalized biamenable.</p>

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Hadwin, Don, and Mehmet Orhon. "A noncommutative theory of Bade functionals." Glasgow Mathematical Journal 33, no. 1 (1991): 73–81. http://dx.doi.org/10.1017/s0017089500008053.

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Since the pioneering work of W. G. Bade [3, 4] a great deal of work has been done on bounded Boolean algebras of projections on a Banach space ([11, XVII.3.XVIII.3], [21, V.3], [16], [6], [12], [13], [14], ]17], [18], [23], [24]). Via the Stone representation space of the Boolean algebra, the theory can be studied through Banach modules over C(K), where K is a compact Hausdorff space. One of the key concepts in the theory is the notion of Bade functionals. If X is a Banach C(K)-module and x ε X, then a Bade functional of x with respect to C(K) is a continuous linear functional α on X such that

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Cao, Huai-Xin, Ji-Rong Lv, and JM Rassias. "Superstability for Generalized Module Left Derivations and Generalized Module Derivations on a Banach Module (I)." Journal of Inequalities and Applications 2009, no. 1 (2009): 718020. http://dx.doi.org/10.1155/2009/718020.

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Dashti, Mahshid, and Sima Soltani Renani. "The retraction of certain banach right modules associated to a character." Mathematica Slovaca 69, no. 4 (2019): 891–900. http://dx.doi.org/10.1515/ms-2017-0260.

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Abstract Let 𝓐 be a Banach algebra and let 𝓜 be a unital Banach algebra. For a homomorphism Φ from 𝓐 into 𝓜, we consider 𝓜 as a Banach right 𝓐-module and investigate when 𝓜 is a retract of 𝓐 with respect to Φ. We also give characterizations of admitting vector-valued invariant Φ-means in terms of projectivity and injectivity. Finally, we apply these results to abstract Segal algebras.

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