The spectral topology of a ring is easily defined, has familiar applications in elementary Banach algebra theory, and appears relevant to abstract Fredholm and stable range theory.
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In the normed space of bounded operators between a pair of normed spaces, the set of operators which are "bounded below" forms the interior of the set of one-one operators. This note is concerned with the extension of this observation to certain spaces of pairs of operators.
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In this paper we consider a subset  of a Banach algebra A (containing all elements of A which have a generalized inverse) and characterize membership in the closure of the invertibles for the elements of Â. Thus our result yields a characterization of the closure of the invertible group for all those Banach algebras A which satisfy  = A. In particular, we prove that  = A when A is a von Neumann algebra. We also derive from our characterization new proofs of previously known results, namely Feldman and Kadison's characterization of the closure of the invertibles in a von Neumann algebra and a more recent characterization of the closure of the invertibles in the bounded linear operators on a Hilbert space.
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Commutativity and continuity conditions for the Moore-Penrose inverse and the "conorm" are established in a C*-algebra; moreover, spectral permanence and B*-properties for the conorm are proved.
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