9 November، 2023

Discussion of a Doctorial Dissertation in the 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 “**

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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**

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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:

- is – generalized derivation.
- is – – derivation and – ternary derivation, when .
- 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

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