Idempotents in Banach algebras II

Abstract

Let A be a complex and unital Banach algebra, and let E denote the collection of idempotents of A. An old paper of J. Zem´anek’s, which  was really the starting point of studies of idempotents in general Banach algebras, exhibits a multitude of results concerning the set E.  More specifically, it was shown that the connected component of E which contains p ∈ E, say Ep, is precisely the set of elements of the  form wpw−1 where w runs through the principal component of the invertible group of A. Following Zem´anek’s work, a considerable  number of papers concerning the form of paths connecting the members of Ep have seen the light of day. In the present paper, we  elaborate on the second part of Zem´anek’s article; by utilizing modern techniques and results, together with a surprisingly general  connection between the spectra of products of idempotents and linear combinations of idempotents, we show that the global algebraic  conditions which characterize central idempotents of A can be replaced by significantly weaker local spectral conditions.