On idempotents and the socle of a Banach algebra

Let Abe a complex and unital Banach algebra and let Ebe the set of idempo- tents of A. In this paper, we investigate properties of idempotents, which belong to the socle of a semisimple Banach algebra as well as how the spectral rank and socle interact with E. We look at what properties an idempotent phas when considering the number of distinct elements in the spectrum of pq, as qvaries in the connected component of Econtaining p. Some examples are provided, which show the existence of Banach alge- bras where it is possible to find idempotents pand qin the same component of Esuch that pqand p+qhave infinite spectra-either countable or uncountable. Under the as- sumption that the socle is a minimal two-sided ideal, we show that (i) two finite-rank idempotents belong to the same component of Eif and only if their ranks coincide; and (ii) the socle is given by the linear span of the component of Econtaining rank one idem- potents. The latter result should be compared with the known fact that, in general, the socle of a semisimple Banach algebra is the linear span of rank one idempotents (poten- tially coming from different components of E).