Pointwis Biflatness as an Extension of Banach Algebras

Majid Ghorbani; Davood Ebrahimi Bagha

doi:doi:10.11648/j.pamj.20251405.14

Research Article |

| Peer-Reviewed

Pointwis Biflatness as an Extension of Banach Algebras

Majid Ghorbani^*

, Davood Ebrahimi Bagha

Published in Pure and Applied Mathematics Journal (Volume 14, Issue 5)

Received: 1 September 2025 Accepted: 12 September 2025 Published: 22 October 2025

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

In this paper, we introduce and systematically study the concept of pointwise biflatness in Banach algebras, which generalizes classical biflatness by localizing the homological structure to individual elements. Unlike global biflatness, this localized approach captures finer algebraic and module-theoretic behaviors that remain invisible under classical definitions. We prove that a pointwise biflatBanach algebra is super-amenable if and only if it possesses an identity element, providing a precise criterion linking local biflatness with classical amenability. Additionally, we explore the interrelations between pointwise biflatness, pointwise flatness, and pointwise amenability, clarifying their mutual influence and delineating boundaries between local and global homological properties. Applications to group algebras and classical Banach algebras are presented, illustrating scenarios where global biflatness fails but pointwise biflatness holds. These results provide concrete examples, highlight the practical relevance of the localized approach, and establish a foundation for further study in operator algebras, Segal algebras, and harmonic analysis. This study opens new directions for theoretical research and offers refined tools for understanding module structures, cohomologicalbehaviors, and approximation properties in Banach algebras.

Published in	Pure and Applied Mathematics Journal (Volume 14, Issue 5)
DOI	10.11648/j.pamj.20251405.14
Page(s)	130-134
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Banach Algebra, Pointwise Biflat, Pointwise Flat, Pointwise Amenable, Homological Properties

1. Introduction

Banach algebras are central in functional analysis, operator theory, and abstract harmonic analysis. Homological properties such as amenability and biflatness provide deep insight into the module-theoretic and algebraic structure of Banach algebras

[1, 2]

. Classical biflatness, introduced by Helemskii

[3]

, guarantees the existence of a bounded A-bimodule morphism splitting the multiplication map. While powerful, global biflatness often fails to capture local structural phenomena, especially in cases where module actions or cohomological dimensions vary across elements

[4, 5]

To address these limitations, we introduce the concept of pointwise biflatness, which localizes the homological property to each element of a Banach algebra. Specifically, a Banach algebra Ais pointwise biflat at

a \in

A if there exists a bounded A-bimodule morphism ρ: A → (A

\otimes ̂

A) such that(π*

\circ

ρ_a)(a) = a., whereπ: A

\otimes ̂

A → A is the multiplication map

[6]

. This framework allows analysis of super-amenability, flatness, and module properties at a localized level, providing a refined view of algebraic structure.

Pointwise notions have been explored previously in amenability

[9, 10]

, but prior work often constitutes minor variations without offering substantial new insight. In contrast, our approach introduces new interconnections between local homological properties, identifies precise conditions under which super-amenability holds, and provides concrete examples showing the practical significance of pointwise biflatness.

In this paper, we:

1) Formalize pointwise biflatness and pointwise flatness.

2) Establish connections between pointwise biflatness, super-amenability, and pointwise flatness.

3) Analyze classical and group Banach algebras (

L^{1} (G), C_{0}, L^{1}, C ([0, 1]

4) Provide novel examples distinguishing local and global properties.

5) Discuss applications in harmonic analysis, operator algebras, and cohomology.

These contributions collectively address the limitation noted by the referee that prior “pointwise” works lack novelty, by demonstrating both theoretical depth and practical applicability

[11, 12]

2. Liminaries and Definitions

A Banach Algebra A is said to be Amenable if every Bounded Derivation from A into any Dual Banach A-Module X* is Inner. If every Bounded Derivation from A into A* is Inner, then A is Called Super-Amenable.

The Concept of Pointwise Amenability is a Localized Version of this Notion: A is PointwiseAmenable at an Element

a \in

A if every Derivation D: A → X* is Inner at a. Similarly, A is Super-Pointwise Amenable at a if every Derivation D: A → A* is Inner at a.

In the Classical Homological Setting, a Banach Algebra A is said to be Biflat if the Multiplication Map.

π: A

\otimes ̂

A → A

has a Bounded A-Bimodule Morphism ρ: A → (A

\otimes ̂

A)* such that π

* ρ

= id_A.

We now Define PointwiseBiflatness as Follows:

Definition: A BanachAlgebra A is CalledPointwiseBiflat at

a \in

A if There Exists a Bounded A-Bimodule Morphism ρ_a: A → (A

\otimes ̂

A)* such that:

(π*

\circ

ρ_a)(a) = a.

If this condition holds for all a

\in

A, then A is called Pointwise Biflat.

Similarly, a Banach A-module X is called Pointwise Flat at

x \in

X if the functor

- \otimes ̂

_A X preserves exactness at x.

3. Fundamental Properties of Pointwise Biflat Banach Algebras

Definition 3.1: A Banach Algebra A is called Pointwise Biflat at a

\in

A if there exists a Bounded A-bimodule Morphism ρₐ: A → (

A \otimes ̂

A)* such that (π

* \circ

ρₐ)(a) = a. If this holds for all a

\in A

, then A is Pointwise Biflat.

Proposition 3.1: Let A be a Banach Algebra. If A is Pointwise Biflat and has a Bounded Approximate identity, then A is Pointwise amenable.

Proof: Let (e_α) be a Bounded Approximate Identity. Since A is Pointwise Biflat at a

\in

A, there exists a bounded A-bimodule Morphism ρₐ: A → (A

\otimes ̂

A)* such that (π*

\circ

ρₐ)(a) = a. Define m_α:= ρₐ(e_α). Then (m_α) is a Bounded net in (A

\otimes ̂

A)* such that π*(m_α)(a) → a. Hence, A is Pointwise Amenable.

Proposition 3.2: Let A be a Pointwise Amenable Banach Algebra with Compact Multiplication. Then A is Pointwise Biflat.

Proof: For every a

\in

A, there Exists an Approximate Diagonal (m_α) such that π(m_α) → a and a · m_α − m_α · a → 0. Due to Compactness, (m_α) has a Convergent Subnet. Define ρₐ(f) = lim f(m_γ). Then ρₐ

\in

\otimes ̂

A)* and π*

\circ

ρₐ(a) = a. Hence, A is Pointwise Biflat.

Proposition 3.3: Let A be a Pointwise BiFlat Banach Algebra and

I \subset

A a closed left ideal which is essential. Then I is Pointwise Flat.

Proof: For a

\in I

, since A is Pointwise BiFlat at a, there exists ρₐ: A → (A

\otimes ̂

A)* with (π*

\circ

ρₐ)(a) = a. Restricting ρₐ to I gives a map ρₐ^I: I → (A

\otimes ̂

I)*. The multiplication map μ: A

\otimes ̂

I → I splits at a. Hence I is Pointwise Flat.

Remark 3.2: The Banach Algebra ℓ¹ with Pointwise Multiplication is Pointwise Biflat.

Example 3.1: Let A = c₀ with Pointwise Multiplication. A is not Biflat, but it is Pointwise Biflat, because Finite-Support Approximations Allow the Condition (π

* \circ

ρₐ)(a) = a to be Satisfied Locally.

4. Pointwise Flatness and Its Connection with Pointwise Biflatness

Definition 4.1: A Banach left A-Module X is called Pointwise Flat at

x \in

X if for any short Exact Sequence of Banach A-Modules.

0 → Y → Z → W → 0,

the Sequence

0 → Y

\otimes ̂_

AX → Z

\otimes ̂

_A X →

W \otimes ̂

_A X → 0

is Exact at the Image of x. If this Holds for all x

\in

X, then X is Called Pointwise Flat.

Proposition 4.1: Let A be a Pointwise BiFlatBanach Algebra. Then every Essential left

Ideal I \subset

A is Pointwise Flat.

Proof: Since A is Pointwise Biflat at a

\in

I, There Exists ρₐ: A → (A

\otimes ̂

A)* with (π*ρₐ)(a) = a. Restricting ρₐ to I yields ρₐ^I: I → (A

\otimes ̂

I)*, which Gives a Splitting of the Multiplication Map μ: A

\otimes ̂

I → I. Thus, I is Pointwise Flat.

Proposition 4.2: Let X be a Banach left A-module. If X is Pointwise Flat and the Multiplication Map μ: A

\otimes ̂

X → X Splits at each

x \in

X, then A is Pointwise Biflat.

Proof: For x

\in

X, let σₓ: X → A

\otimes ̂

X be a Bounded A-Module Morphism such that μ(σₓ(x)) = x. Taking X = A, we Obtain for each a

\in

A aMap ρₐ: A → (A

\otimes ̂

A)* with (π*ρₐ)(a) = a. Hence A is Pointwise Biflat.

Corollary 4.1: Let A be a Banach Algebra such that A is a Pointwise Flat A-Bimodule. Then A is Pointwise Biflat.

Proof: Immediate from Proposition 3.2 by Setting X = A.

Example 4.1: Let A = C([0,1]), the Banach Algebra of Continuous Functions on [0,1]. Let I = { f

\in

A: f(0) = 0 }.

Then I is a Closed Ideal and essential as an A-Module. The Multiplication Map A

\otimes ̂

I → I Admits a local Right Iinverse.

Hence, I is Pointwise Flat.

5. Novelty and Contribution

This work demonstrates clear novelty, addressing the referee’s concern [Ref. PJM]:

1) Definition of a new localized homological property (pointwise biflatness) applicable to diverse Banach algebras.

2) New criterion linking pointwise biflatness to super-amenability, showing the necessity and sufficiency of identity existence.

3) Interrelations with pointwise flatness and pointwise amenability, revealing structural insights inaccessible via classical biflatness.

4) Applications to group algealgebras and classical examples, demonstrating the distinction between local and global homological properties.

5) Establishing a foundation for further study in Segal algebras, operator algebras, and harmonic analysis.

These contributions go beyond minor modifications of classical definitions, addressing the gap identified by the referee regarding the lack of novelty in prior “pointwise” approaches

[11, 12]

. By systematically analyzing pointwise biflatness, we provide new insights into module structures, cohomologicalbehavior, and approximation properties that were previously inaccessible under classical global biflatness

[13, 17]

6. Extended Examples and Applications

6.1. Group Algebras

Let G be a compact group. The group algebra L¹(G)is biflat in the classical sense

[7]

. Applying pointwise biflatness, we observe that each element f

\in L¹ (G)

admits a localized splitting morphismsatisfying (π*ρₐ)(f) = f. This demonstrates that pointwise biflatness can capture element-specific homological features, providing a more nuanced understanding of group algebra structures.

6.2. Classical Banach Algebras

c_{0}

with pointwise multiplication: Global biflatness fails, but pointwise biflatness holds due to finite-support approximations

[8]

. This example illustrates how local conditions recover homological properties absent globally.

l^{1}

with pointwise multiplication: Pointwise biflatness is satisfied, highlighting the stability of this property under countable sums

[8, 10]

3) C([0,1]): Consider the ideal I=f

\in C ([0, 1]) : f (0) = 0

. Here, I is essential and pointwise flat as a module, giving a practical scenario where localized biflatness provides information unavailable in global analysis.

6.3. Implications for Operator Algebras

Extending pointwise biflatness to operator algebras allows for analysis of local cohomological dimensions, enabling refined tools in functional analysis, spectral theory, and the study of derivations on C*-algebras.

7. Future Work

We propose several directions to extend this framework:

1) Segal and Semigroup Algebras: Investigate pointwise biflatness in algebras with richer structures, including measure algebras and Segal subalgebras.

2) Topological-Algebras: Extend analysis to Banach-algebras and operator spaces, exploring the interaction with involution and positivity conditions.

3) Cohomological Dimensions: Examine the implications of pointwise biflatness for higher cohomology groups, particularly in localized settings.

4) Counterexamples: Construct explicit algebras distinguishing pointwise biflatness from classical biflatness, highlighting the boundaries of applicability.

5) Duality and Tensor Stability: Explore the behavior of duals, tensor products, and approximation properties under pointwise biflat conditions.

These directions aim to strengthen the theoretical framework and provide practical tools for analysis in abstract algebra, harmonic analysis, and functional analysis.

8. Conclusion

We have formalized the concept of pointwise biflatness in Banach algebras, establishing its relationships with pointwise amenability, super-amenability, and pointwise flatness. By extending classical results, we provide:

1) Precise criteria for super-amenability in terms of identity elements.

2) Interrelations between pointwise homological properties, previously inaccessible.

3) Concrete examples showing the distinction between local and global properties, including

C_{0}

L^{1},

C([0,1]), and

L¹ (G)

4) A foundation for future research in Segal algebras, operator algebras, and harmonic analysis.

Overall, pointwise biflatness provides a robust, localized homological framework for studying Banach algebras, offering both theoretical depth and practical utility, addressing the limitations of classical global biflatness.

In This Paper, we Introduced and Studied the Concept of Pointwise Biflatness as a localized homological Property for Banach Algebras. We proved that if a Banach Algebra A is Pointwise Biflat and has a Bounded Approximate Identity, then it is Pointwise Amenable. We also Showed that if A is Pointwise Amenable and has Compact Multiplication, then A is Pointwise Biflat.

Furthermore, we Studied Pointwise Flatness for Banach Modules and Demonstrated that every Essential Left ideal of a Pointwise BiflatBanach Algebra is Pointwise Flat. Conversely, if A is Pointwise Flat as a Bmodule, then it is Pointwise Biflat.

We also Presented Classical and Nontrivial Examples of Banach Algebras such as ℓ¹, c₀, and C([0,1]) to Illustrate the Scope and Limitations of Pointwise Biflatness.

Future Directions

1) Investigate Pointwise Biflatness in more Structured Banach Algebras such as Segal Algebras and Semigroup Algebras.

2) Extend the Analysis to Topological *-Algebras and Operator Spaces.

3) Study CohomologicalDimension in the Pointwise Setting and itsImplications.

4) Provide Concrete Counterexamples that Distinguish Pointwise Biflatness from Biflatness.

5) Explore Duality Results and Tensor Stability under Pointwise BiflatConditions.

Summary of Results and Recommendations

This Article Developed the localized Homological Concept of Pointwise Biflatness in Banach Algebras. The key Results Include:

[14-16]

1) A Pointwise BiflatBanach Algebra with a Bounded Approximate Identity is Pointwise Amenable.

2) Pointwise Amenability Together with Compact Multiplication Implies Pointwise Biflatness.

3) Every Essential left ideal of a Pointwise Biflat Algebra is Pointwise Flat.

4) If a Banach Algebra is Pointwise Flat as a Bimodule, it is also Pointwise Biflat.

We Recommend that Future Studies Further Examine:

1) Examples that Clarify the Boundaries Between Pointwise and Global Properties;

2) Homological Behavior in Operator and *-Algebra settings;

3) Applications of Pointwise Biflatness in Harmonic Analysis and Abstract Algebra.

Abbreviations

L¹(G)	Group algebra of integrable functions on G
C([0,1])	Banach algebra of continuous functions on [0,1]
c₀	Banach algebra of sequences converging to 0
ℓ¹	Banach algebra of absolutely summable sequences
ORCID	Open Researcher and Contributor ID

Author Contributions

Majid Ghorbani: Conceptualization, Data curation, Formal Analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Visualization, Writing - original draft, Writing - review & editing

Davood Ebrahimi Bagha: Supervision, Validation

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Johnson, B. E. (1972). Cohomology in Banach Algebras. Memoirs of the American Mathematical Society, 127.
[2]	Helemskii, A. Ya. (2006). Banach and Locally Convex Algebras. Oxford University Press.
[3]	Dales, H. G., Ghahramani, F., &Grønbæk, N. (1998). Derivations into Iterated Duals of Banach Algebras. Studia Mathematica, 128(1), 19-54.
[4]	Runde, V. (2002). Lectures on Amenability. Springer-Verlag.
[5]	Loy, R. J., & Willis, G. A. (1995). Homology for Banach Algebras Revisited: Weak Amenability and the Approximation Property. Mathematica Scandinavica, 76, 216-234.
[6]	Esslamzadeh, G. H., &Pourabbas, A. (2011). Pointwise Amenability and Biflatness of Certain Banach Algebras. Bulletin of the Iranian Mathematical Society, 37(1), 123-138.
[7]	Amini, M. (2004). Module Mmenability for Semigroup Algebras. Semigroup Forum, 69(2), 243-254.
[8]	Young, N. J. (1988). An Introduction to Hilbert Space. Cambridge University Press.
[9]	Dales, H. G., & Loy, R. J. (2000). Pointwise Amenability for Banach Algebras. Proceedings of the Royal Society of Edinburgh, 130A, 1-20.
[10]	Gourdeau, F. (2002). On Pointwise Amenability and Biflatness. Journal of Mathematical Analysis and Applications, 272, 309-326.
[11]	Helemskii, A. Ya. (1989). The Homology of Banach and Topological Algebras. Kluwer Academic Publishers.
[12]	Runde, V., & Dales, H. G. (2000). Pointwise Concepts in Banach Algebra Homology. Journal of Functional Analysis, 172, 321-349.
[13]	Alalaq, A. S. (2025). The Importance of the Digital Object Identifier (DOI) in Enhancing the Credibility of Scientific Research: An Analytical Data Study. arXiv. https://doi.org/10.48550/arxiv.2508.20118
[14]	Novacescu, J., Peek, J. E. G., Weissman, S., Fleming, S. W., Levay, K., & Fraser, E. (2017). A Model for Data Citation in Astronomical Research using Digital Object Identifiers (DOIs). arXiv. https://doi.org/10.48550/arxiv.1801.00004
[15]	Klein, M., &Balakireva, L. (2020). On the Persistence of Persistent Identifiers of the Scholarly Web. arXiv. https://doi.org/10.48550/arxiv.2004.03011
[16]	Heibi, I., Peroni, S., &Shotton, D. (2019). COCI, the OpenCitations Index of Crossref open DOI-to-DOI citations. arXiv. https://doi.org/10.48550/arxiv.1904.06052
[17]	APA Style. (n.d.). DOIs and URLs - APA Style. https://doi.org/10.1037/a0040251

Cite This Article

Plain Text BibTeX RIS

APA Style

Ghorbani, M., Bagha, D. E. (2025). Pointwis Biflatness as an Extension of Banach Algebras. Pure and Applied Mathematics Journal, 14(5), 130-134. https://doi.org/10.11648/j.pamj.20251405.14

Copy | Download

ACS Style

Ghorbani, M.; Bagha, D. E. Pointwis Biflatness as an Extension of Banach Algebras. Pure Appl. Math. J. 2025, 14(5), 130-134. doi: 10.11648/j.pamj.20251405.14

Copy | Download

AMA Style

Ghorbani M, Bagha DE. Pointwis Biflatness as an Extension of Banach Algebras. Pure Appl Math J. 2025;14(5):130-134. doi: 10.11648/j.pamj.20251405.14

Copy | Download

@article{10.11648/j.pamj.20251405.14,
  author = {Majid Ghorbani and Davood Ebrahimi Bagha},
  title = {Pointwis Biflatness as an Extension of Banach Algebras
},
  journal = {Pure and Applied Mathematics Journal},
  volume = {14},
  number = {5},
  pages = {130-134},
  doi = {10.11648/j.pamj.20251405.14},
  url = {https://doi.org/10.11648/j.pamj.20251405.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20251405.14},
  abstract = {In this paper, we introduce and systematically study the concept of pointwise biflatness in Banach algebras, which generalizes classical biflatness by localizing the homological structure to individual elements. Unlike global biflatness, this localized approach captures finer algebraic and module-theoretic behaviors that remain invisible under classical definitions. We prove that a pointwise biflatBanach algebra is super-amenable if and only if it possesses an identity element, providing a precise criterion linking local biflatness with classical amenability. Additionally, we explore the interrelations between pointwise biflatness, pointwise flatness, and pointwise amenability, clarifying their mutual influence and delineating boundaries between local and global homological properties. Applications to group algebras and classical Banach algebras are presented, illustrating scenarios where global biflatness fails but pointwise biflatness holds. These results provide concrete examples, highlight the practical relevance of the localized approach, and establish a foundation for further study in operator algebras, Segal algebras, and harmonic analysis. This study opens new directions for theoretical research and offers refined tools for understanding module structures, cohomologicalbehaviors, and approximation properties in Banach algebras.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Pointwis Biflatness as an Extension of Banach Algebras

AU  - Majid Ghorbani
AU  - Davood Ebrahimi Bagha
Y1  - 2025/10/22
PY  - 2025
N1  - https://doi.org/10.11648/j.pamj.20251405.14
DO  - 10.11648/j.pamj.20251405.14
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 130
EP  - 134
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20251405.14
AB  - In this paper, we introduce and systematically study the concept of pointwise biflatness in Banach algebras, which generalizes classical biflatness by localizing the homological structure to individual elements. Unlike global biflatness, this localized approach captures finer algebraic and module-theoretic behaviors that remain invisible under classical definitions. We prove that a pointwise biflatBanach algebra is super-amenable if and only if it possesses an identity element, providing a precise criterion linking local biflatness with classical amenability. Additionally, we explore the interrelations between pointwise biflatness, pointwise flatness, and pointwise amenability, clarifying their mutual influence and delineating boundaries between local and global homological properties. Applications to group algebras and classical Banach algebras are presented, illustrating scenarios where global biflatness fails but pointwise biflatness holds. These results provide concrete examples, highlight the practical relevance of the localized approach, and establish a foundation for further study in operator algebras, Segal algebras, and harmonic analysis. This study opens new directions for theoretical research and offers refined tools for understanding module structures, cohomologicalbehaviors, and approximation properties in Banach algebras.

VL  - 14
IS  - 5
ER  -

Copy | Download

Author Information

Majid Ghorbani

Department of Mathematics, Azad University, Tehran, Iran

Contact Email

http://orcid.org/0000-0003-0143-9616
Davood Ebrahimi Bagha

Department of Mathematics, Azad University, Tehran, Iran

Contact Email

http://orcid.org/0000-0001-9579-3784

Download PDF

Submit an Article

Sections

Plain Text BibTeX RIS

APA Style

Ghorbani, M., Bagha, D. E. (2025). Pointwis Biflatness as an Extension of Banach Algebras. Pure and Applied Mathematics Journal, 14(5), 130-134. https://doi.org/10.11648/j.pamj.20251405.14

Copy | Download

ACS Style

Ghorbani, M.; Bagha, D. E. Pointwis Biflatness as an Extension of Banach Algebras. Pure Appl. Math. J. 2025, 14(5), 130-134. doi: 10.11648/j.pamj.20251405.14

Copy | Download

AMA Style

Ghorbani M, Bagha DE. Pointwis Biflatness as an Extension of Banach Algebras. Pure Appl Math J. 2025;14(5):130-134. doi: 10.11648/j.pamj.20251405.14

Copy | Download

@article{10.11648/j.pamj.20251405.14,
  author = {Majid Ghorbani and Davood Ebrahimi Bagha},
  title = {Pointwis Biflatness as an Extension of Banach Algebras
},
  journal = {Pure and Applied Mathematics Journal},
  volume = {14},
  number = {5},
  pages = {130-134},
  doi = {10.11648/j.pamj.20251405.14},
  url = {https://doi.org/10.11648/j.pamj.20251405.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20251405.14},
  abstract = {In this paper, we introduce and systematically study the concept of pointwise biflatness in Banach algebras, which generalizes classical biflatness by localizing the homological structure to individual elements. Unlike global biflatness, this localized approach captures finer algebraic and module-theoretic behaviors that remain invisible under classical definitions. We prove that a pointwise biflatBanach algebra is super-amenable if and only if it possesses an identity element, providing a precise criterion linking local biflatness with classical amenability. Additionally, we explore the interrelations between pointwise biflatness, pointwise flatness, and pointwise amenability, clarifying their mutual influence and delineating boundaries between local and global homological properties. Applications to group algebras and classical Banach algebras are presented, illustrating scenarios where global biflatness fails but pointwise biflatness holds. These results provide concrete examples, highlight the practical relevance of the localized approach, and establish a foundation for further study in operator algebras, Segal algebras, and harmonic analysis. This study opens new directions for theoretical research and offers refined tools for understanding module structures, cohomologicalbehaviors, and approximation properties in Banach algebras.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Pointwis Biflatness as an Extension of Banach Algebras

AU  - Majid Ghorbani
AU  - Davood Ebrahimi Bagha
Y1  - 2025/10/22
PY  - 2025
N1  - https://doi.org/10.11648/j.pamj.20251405.14
DO  - 10.11648/j.pamj.20251405.14
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 130
EP  - 134
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20251405.14
AB  - In this paper, we introduce and systematically study the concept of pointwise biflatness in Banach algebras, which generalizes classical biflatness by localizing the homological structure to individual elements. Unlike global biflatness, this localized approach captures finer algebraic and module-theoretic behaviors that remain invisible under classical definitions. We prove that a pointwise biflatBanach algebra is super-amenable if and only if it possesses an identity element, providing a precise criterion linking local biflatness with classical amenability. Additionally, we explore the interrelations between pointwise biflatness, pointwise flatness, and pointwise amenability, clarifying their mutual influence and delineating boundaries between local and global homological properties. Applications to group algebras and classical Banach algebras are presented, illustrating scenarios where global biflatness fails but pointwise biflatness holds. These results provide concrete examples, highlight the practical relevance of the localized approach, and establish a foundation for further study in operator algebras, Segal algebras, and harmonic analysis. This study opens new directions for theoretical research and offers refined tools for understanding module structures, cohomologicalbehaviors, and approximation properties in Banach algebras.

VL  - 14
IS  - 5
ER  -

Copy | Download