The generalized Fredholm elements in a semisimple Banach algebra

Let A be a unital semisimple Banach algebra. Denote the set of the generalized Fredholm elements in A by Φ _g(A). In this paper, we study the perturbations of the generalized Fredholm elements and the spectral mapping theorem of the generalized Fredholm spectrum. Furthermore, for a∈ Φ _g(A) , the conditions that f(a) is also a generalized Fredholm element are investigated, where f is a complex-valued function analytic on a neighborhood of σ(a). In addition, the topological structure of Φ _g(A) are discussed. As an application, the socle of a primitive C^∗-algebra is characterized by the generalized Fredholm elements.