Various topological forms of Von Neumann regularity in Banach algebras

We study topological von Neumann regularity and principal von Neumann regularity of Banach algebras. Our main objective is comparing these two types of Banach algebras and some other known Banach algebras with one another. In particular, we show that the class of topologically von Neumann regular Banach algebras contains all $C^*$-algebras, group algebras of compact abelian groups and certain weakly amenable Banach algebras while it excludes measure algebras, of certain locally compact Abelian groups. Moreover, we show that in a unital amenable Banach algebra, principal regularity implies topological regularity. Finally, we use topological regularity to obtain some information about hereditary $C^*$-subalgebras of a given $C^*$-algebra.