Consistency of Fredholmness Under Products in Semisimple Banach Algebras

Let A be a unital infinite dimensional semisimple Banach algebra and Φ(A) be the set of Fredholm elements in A. An element a∈A is called Φ(A)-consistent provided that ab∈Φ(A) if and only if ba∈Φ(A) for every b∈A. We first characterize the Φ(A)-consistent elements and show that the set of such elements forms an upper semiregularity. Building on this, we introduce the consistent Fredholm spectrum, establish its spectral mapping theorem, and obtain a characterization for algebraic elements in A. As an application, we characterize the stability of Φ(A)-consistent elements in terms of nullity and defect in primitive c∗ - algebras.