ZERO SUMS OF IDEMPOTENTS IN BANACH-ALGEBRAS

The problem treated in this paper is the following. Let p1,...,p(k) be idempotents in a Banach algebra B, and assume P1+...+P(k)=0. Does it follow that p(j)=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.