Idempotents in Banach algebras II

Let A be a complex and unital Banach algebra, and let E denote the collection of idempotents of A. An old paper of J. Zemánek’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 E_p, 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 Zemánek’s work, a considerable number of papers concerning the form of paths connecting the members of E_p have seen the light of day. In the present paper, we elaborate on the second part of Zemánek’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.