The Relative (Co)homology Theory through Operator Algebras

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 <img src=image/13426521_01.gif>. The definitions of cyclic, simplicial, and dihedral cohomology group of <img src=image/13426521_01.gif> will be introduced. We presented the definition of <img src=image/13426521_14.gif>-relative dihedral cohomology group that is given by: <img src=image/13426521_02.gif>, and we will show that the relation between dihedral and <img src=image/13426521_14.gif>-relative dihedral cohomology group <img src=image/13426521_11.gif> <img src=image/13426521_12.gif> <img src=image/13426521_13.gif> can be obtained from the sequence <img src=image/13426521_03.gif>. 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 (<img src=image/13426521_04.gif> and <img src=image/13426521_05.gif>) of <img src=image/13426521_01.gif> will be proved as the sequence <img src=image/13426521_06.gif>. 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 <img src=image/13426521_07.gif> and <img src=image/13426521_08.gif>, and proved the Connes-Tsygan exact sequence that relates the relative cyclic and dihedral (co)homology of <img src=image/13426521_01.gif>. We proved the Mayer-Vietoris sequence of <img src=image/13426521_04.gif> in a new form in the Banach B-relative dihedral cohomology: <img src=image/13426521_09.gif> <img src=image/13426521_10.gif>. 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.