Abstract: The cohomology theory is discussed as the following: Triangular Banach algebras second-order cohomology groups are researched in [10]. Factorization and bounded Hochschild cohomology of matrix-like Banach algebras are investigated in [14], while the group cohomology of triangular Banach algebra is addressed in [5]. In [7], projective modules in quantized homology are introduced. [6] Studies projective Hilbert modules over operator algebras. C^*-algebra projective homological classification is investigated. Using aspects of quantum theory, biprojective algebras, homological dimensions, and current advances in Banach homology theory are addressed in [2]. In [8], the studies the disappearance of the third simplicial cohomology group for specific classes of C^*-algebra. In [95], an approximate identity for ideals of Segal algebras on a compact group and homology theory are examined.