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 A. The definitions of cyclic, simplicial, and dihedral cohomology group of A will be introduced. We presented the definition of ℬ-relative dihedral cohomology group that is given by: HDn[(A,B),X]=Zn[(A,B),X]/Bn[(A,B),X], and we willshow that the relation between dihedral and B-relative dihedral cohomology group 0 →HD0[(A,B),X]→HD0(A,X)→HD0(A/B. X) →HD1[(A,B),X]→HD1(A,X)→ ⋯ →HDn[(A,B),X] →HDn(A,X)→HDn(A/B,X)→HDn+1[(A,B),X) →⋯ can be obtained from the sequence 0→CDn[(A,B),X] →CDn(A,X) →CDn(A/B,X) →0. 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 (HDn[(A,B),X] and HCn[(A,B),X] of A will be proved as the sequence →−HDn[(A,B),X]i* →HCn[(A,B),X]j*→HDn[(A,B),X]→-HDn+1[(A,B),X] →⋯. 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 tr∗:HD*(Mr(B),Mr(M)) →HD*(B,M) and inc∗: HD*(B,M→HD* HD*(Mr(B),Mr(M)), and proved the Connes-Tsygan exact sequence that relates the relative cyclic and dihedral (co)homology of A. We proved the Mayer-Vietoris sequence of HDn[(A,B),X] in a new form in the Banach B-relative dihedral cohomology: (Formula Presented). 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. © 2022 by authors, all rights reserved.