Strong Topological Regularity and Weak Regularity of Banach Algebras

In this article we study two different generalizations of von Neumann regularity, namely strong topological regularity and weak regularity, in the Banach algebra context. We show that both are hereditary properties and under certain assumptions, weak regularity implies strong topological regularity. Then we consider strong topological regularity of certain concrete algebras. Moreover we obtain the following non-commutative analog of a result of Kaplansky. A bounded operator T on a Banach space X whose point spectrum ?p(T) contains a nonzero complex number, is weakly regular.