College of Computer Science and Mathematics

Discussion of a Doctorial Dissertation  in the College of Computer Science and Mathematics – Department of Mathematics Sciences entitled:

“Applications of Module Extension Banach Algebras “

 

In continuation of the scientific research movement and with the follow-up and presence of the Dean of the College of Computer Science and Mathematics

Prof. Dr. Duha Bashir Abdullah The Esteemed

It was discussed in the discussion room at the Faculty of Computer Science and Mathematics at the  University of Mosul on Thursday  9-11-2023

Doctorial Dissertation by student Lamia Kader Ismail Khalil

, under the supervision of Prof. Dr. Dr. Amir Abdulillah Mohammed Hussin

                  

This thesis is based on similar to the study the properties of the linear mappings on Banach algebras by many researchers, we study and generalized some types of the derivations on module extension Banach algebra, depending on the two homomorphisms that effects on module extension Banach algebras.

Let  be module extension Banach –algebra and  be a Banach -bimodule, and let ,  and  be continuous homomorphisms on  to . We proved the following:

If  is a module extension Banach algebra with zero product and every  and  are essential, then the continuous linear mapping   is -anti derivation at zero, if and only if there exist a -anti derivation  and  such that  .

And we proved a continuous – local ternary derivation from unital module extension Banach -algebra into their periodical duals are symmetrical mapping and  -ternary derivation.

And also, if the continuous linear mapping on unital  is  -ternary derivation at the unite element (at zero), then:

1. is – generalized derivation.
2. is – – derivation and – ternary derivation, when .
3. is – ternary derivation.

In addition to that we proved, if  is a symmetric amenability such that  is weakly sequentially complete and  is essential, and the continuous linear mapping  from  into  is -Jorden derivation at zero, then there exists -derivation  and also there exists homomorphism , identifiers from  into  such that . We obtained similar results of the continuous linear mappings that are  – – Jorden derivations at zero on module extension Banach -algebras.

Finally, we proved each – local higher derivation on a symmetric amenability module extension Banach algebra is – higher derivation. In addition to that, each – – anti local higher derivation on a symmetric amenability module extension Banach -algebra is – -higher derivation.

 

The scientific committee included the following members:

prof.Dr. nizar hamdun shakr  – Chairman

prof. Dr. Ammar Siddiq Mahmoud – Member

prof. Dr.  Dr. Nada Yassin Qasim – Member

prof. Dr. husam qasim muhamad  – member

Ass. Prof. Dr. narin sabih muhamad  – Member

prof.Dr. Amir Abdulillah Mohammed Hussin – Member and supervisor