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.
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