Almost Generalized Derivation on Banach Algebras

Round 1

Reviewer 1 Report

In the paper under consideration the authors study Ulam's Problem for approximate homomorphisms in the case of generalized derivations. In particular they explore the stability of generalized derivations in the context of Banach algebras. They have succeeded to formulate and prove new theorems for approximate generalized derivations on Banach algebras. The present work applies to C*-algebras.

Notwithstanding the fact that the paper is really nice and interesting to the mathematicians working in this domain, I feel that some minor revision could be incorporated to increase the visibility and usefulness to the paper to a broader community of readers. For this to be achieved, it would be really useful for the authors to possibly add to the introduction some further information (that is more results in the domain - both old and new) which introduce the readers to the domain and give an overview of what has been achieved over the years. Some very relevant sources that could help the authors are the following:

S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, 2011.

S. -M. Jung et al, On the stability of the linear functional equation in a single variable on complete metric groups, Journal of Global Optimization, 59(2014), 165–171.

S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, 2002.

Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer, 2009.

C. Park et al, Hom-derivations in C*-ternary algebras, Acta Mathematica Sinica, English Series, 36(9)(2020), 1025-1038.

C. Park et al, Additive functional equations and partial multipliers in  C*-algebras, Revista de la Real Academia de Ciencias Exactas, Serie A. Matemáticas, 113(3)(2019), 2261-2275.

On the basis of the above, I recommend the paper for publication in this Journal.

Author Response

As suggested by the reviewer, at the end of page 1 of this paper, the following sentence was inserted and references were cited:    ``The reader is referred to book and journal for many information of stability problem with a large variety of applications  (for example, [....])."  Of course, the books and papers cited in the references were added.

Reviewer 2 Report

 

See the attached file.

 

Comments for author File: Comments.pdf

Author Response

It  has been revised as the reviewer indicates. 

Reviewer 3 Report

In this paper the author has studied "Almost generalized derivation on Banach algebras.
Authors introduced a generalized derivation and studied its stability via some theorems and lemmas. My comments are appended below point-wise:

1. The underlying concept studied in this paper is outstanding and this paper can be treated as a path breaking work in the coming years. 

2. Results computed in this paper are correct and presented in a very clear fashion. Findings are stated very clearly.

3. Reference list is complete.

Based on the above mentioned facts I strongly recommend publication of this paper in its present form in the journal of Mathematics.

Author Response

The authors would like to thank the referees for giving useful suggestions and for for your time and efforts.