The generalized Fredholm elements in a semisimple Banach algebra (Q2146317)

First the authors study the perturbations of the generalized Fredholm elements on a unital semisimple Banach algebra and the spectral mapping theorem of the generalized Fredholm spectrum. Then they investigate the condition that \(f(a)\) is also a generalized Fredholm element whenever \(a\) is a generalized Fredholm element and \(f\) is a complex-valued function analytic on a neighborhood of the spectrum of \(a\). As an application, they prove that the socle of a primitive C\(^\ast\)-algebra is characterized by the generalized Fredholm elements.