Zero sums of idempotents in Banach algebras

The problem treated in this paper is the following. Let p1,..., pkbe idempotents in a Banach algebra B, and assume p1+...+pk=0. Does it follow that pj=0, j=1,..., k? For important classes of Banach algebras the answer turns out to be positive; in general, however, it is negative. A counterexample is given involving five nonzero bounded projections on infinite-dimensional separable Hilbert space. The number five is critical here: in Banach algebras nontrivial zero sums of four idempotents are impossible. In a purely algebraic context (no norm), the situation is different. There the critical number is four