Zero sums of idempotents in Banach algebras (Q1330846)

The paper deals with the following question: if \(p_ 1,\dots, p_ k\) are idempotents in a Banach algebra \(B\) such that \(p_ 1+ \dots+ p_ k=0\), does it imply that all \(p_ j=0\)? The authors show that the answer is in the affirmative for commutative algebras, polynomial-identity algebras, algebras generated by all compact operators on \(L^ 2(\mathbb{T})\), \(\mathbb{T}=\{ z\in\mathbb{C}\): \(| z|=1\}\), and some others. In general, however, the answer is in the negative. A counterexample is given involving five nonzero bounded projections on infinite-dimensional separable Hilbert space. It is also shown that the number 5 is sharp: nontrivial zero sums of four idempotents are impossible in all Banach algebras. (In a purely ring context (no norm), it has been known for some time that the crucial number is four -- cf. references in the paper).