The triviality of dihedral cohomology for operator algebras

This article delves into algebraic topology, specifically (co)homology theory, which is essential in various mathematical fields. It explores different types of (co)homology groups such as Hochschild, cyclic, reflexive, and dihedral, focusing on dihedral cohomology generated by the dihedral group operating on simplicial complexes. The text discusses when dihedral and reflexive cohomology vanishes and provides examples, along with proving a homomorphism between dihedral cohomology groups. Mathematical concepts like Banach spaces, $C^{*}$-algebras, and tensor products are elucidated to understand the algebraic structures involved. Theorems and proofs establish the relationships and properties of these cohomology groups, culminating in a canonical isomorphism theorem for the product of two algebras. Overall, the article aims to provide a comprehensive exploration of dihedral cohomology and its interplay with other algebraic structures, offering insights into its applications and theoretical foundations. Additionally, it studies the (co)homology theory of $C^{*}$-algebras, focusing on the triviality of cohomology groups of operator algebras and discovering the canonical isomorphism between any two unital $K$ -algebras $A$ and $A{}^{\prime }$: $H{D}_n\left(A\times A^{\prime }\right)\cong H{D}_n\left(A\right)\oplus H{D}_n\left(A^{\prime }\right)$.