On the decomposition of the identity into a sum of idempotents (Q2754845)

Let \(q_1+\ldots +q_n=e\) be a decomposition of the identity element of an algebra \(A\) into a sum of idempotent elements. The authors study the following problem. NEWLINENEWLINENEWLINEIs it always true that \(q_iq_j=0\) for all \(i\neq j\)? NEWLINENEWLINENEWLINEThe answer is positive if \(n=3\) and negative (even for a Banach algebra \(A\)) if \(n\geq 5\). For \(n=4\) the answer is, in general, negative. However, it is positive if \(A\) is a topological complex algebra with continuous inverse.