Commutativity of the Restricted connected hull in Banach algebras

The restricted topology is defined to investigate sets between the spectrum and its hull that arise by filling the holes. In this paper, we study this topology in the context of Banach algebras. We explore the properties of the associated connected hull that are in common or in contrast with the well-known notions of spectrum and exponential spectrum. Further, we deduce sufficient conditions for the commutativity of the connected hull.