### Journals Information

**
Mathematics and Statistics Vol. 10(3), pp. 468 - 476 DOI: 10.13189/ms.2022.100302 Reprint (PDF) (416Kb) **

## The Relative (Co)homology Theory through Operator Algebras

**M. Kozae ^{1}, Samar A. Abo Quota ^{2}^{,*}, Alaa H. N. ^{2}**

^{1}Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt

^{2}Department of Mathematics, Faculty of Science, Aswan University, Aswan, Egypt

**ABSTRACT**

This paper introduces a new idea in the unital involutive Banach algebras and its closed subset. This paper aims to study the cohomology theory of operator algebra. We will study the Banach algebra as an applied example of operator algebra, and the Banach algebra will be denoted by . The definitions of cyclic, simplicial, and dihedral cohomology group of will be introduced. We presented the definition of -relative dihedral cohomology group that is given by: , and we will show that the relation between dihedral and -relative dihedral cohomology group can be obtained from the sequence . Among the principal results that we will explain is the study of some theorems in the relative dihedral cohomology of Banach algebra as a Connes-Tsygan exact sequence, since the relation between the relative Banach dihedral and cyclic cohomology group ( and ) of will be proved as the sequence . Also, we studied and proved some basic notations in the relative cohomology of Banach algebra with unity and defined its properties. So, we showed the Morita invariance theorem in a relative case with maps and , and proved the Connes-Tsygan exact sequence that relates the relative cyclic and dihedral (co)homology of . We proved the Mayer-Vietoris sequence of in a new form in the Banach B-relative dihedral cohomology: . It should be borne in mind that the study of the cohomology theory of operator algebra is concerned with studying the spread of Covid 19.

**KEYWORDS**

Cyclic Homology, Dihedral Cohomology, Exact Sequence, Mayer-Vietoris Sequence, Operator Algebra, Relative Cohomology

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] M. Kozae , Samar A. Abo Quota , Alaa H. N. , "The Relative (Co)homology Theory through Operator Algebras," Mathematics and Statistics, Vol. 10, No. 3, pp. 468 - 476, 2022. DOI: 10.13189/ms.2022.100302.

(b). APA Format:

M. Kozae , Samar A. Abo Quota , Alaa H. N. (2022). The Relative (Co)homology Theory through Operator Algebras. Mathematics and Statistics, 10(3), 468 - 476. DOI: 10.13189/ms.2022.100302.