On the Homology Theory of Operator Algebras

We investigate the cyclic homology and free resolution effect of a commutative unital Banach algebra. Using the free resolution operator, we define the relative cyclic homology of commutative Banach algebras. Lemmas and theorems of this investigation are studied and proved. Finally, the relation between cyclic homology and relative cyclic homology of Banach algebra is deduced. 1. Introduction Many years ago, cyclic homology has been introduced by Connes and Tsygan and defined on suitable categories of algebras, as the homology of a natural chain complex and the target of a natural Chern character from topological (or algebraic) K-Theory. In order to extend the classical theory of the Chern character to the noncommutative setting, Connes [1] and Tsygan [2] have developed the cyclic homology of associative algebras. Recently, there has been increasing interest in general algebraic structures than associative algebras, characterized by the presence of several algebraic operations. Such structures appear, for example, in homotopy theory [3, 4] and topological field theory [5]. Brylinski and Nistor [6] have extended Conne’s computation of the cyclic cohomology groups of smooth algebras arising from foliations with separated graphs and explained some results of Atiyah and Segal on orbifold Euler characteristic in the setting of cyclic homology. Kazhdan [7] studied Hochschild and cyclic homology of finite type algebras using abelian stratifications of their primitive ideal spectrum. Victor Nistor [8] has studied associative -summable quasi homomorphism’s and -summable extensions elements in a bivariant cyclic cohomology group defined by Connes, and showed that this generalizes his character on K-homology; furthermore, he studied the properties of this character and showed that it is compatible with analytic index. Results of Connes [1] have led much research interest into the computation of cyclic (co)homology groups in recent years (see, [4, 6, 9–13]). A promising approach to the calculation of cyclic cohomology groups is to break it down by making use of extensions of Banach algebras; this is a standard device in the study of various properties of Banach algebras. The Banach cyclic (co)homology of Banach algebra has been studied by Christensen and Sinclair [3], Helemskii [4, 9], among others. The dihedral cohomology in Banach category and its relation with the cyclic cohomology, the triviality, and nontriviality of dihedral cohomology groups of some classes of operator algebras have been studied [14]. Suppose that and be commutative unital Banach algebra