On the minimum degree of polynomial connections between projections in a Banach algebra (Q1568121)

The components of the set of idempotents in a Banach algebra \(A\) are arcwise connected, and any two points \(p\) and \(q\) can be connected by an arc defined by a polynomial [\textit{J. Esterle}, Bull. Lond. Math. Soc. 15, 253-254 (1983; Zbl 0517.46034)]. The present paper deals with the question how low the degree of the connecting polynomial can be chosen. It is shown that degree 3 is sufficient, if \(A\) is finite-dimensional and semisimple, or if -- supposed \(A\) is the Banach algebra of all bounded linear operators on a Banach space \(E\) -- \(\text{Im }p\oplus \text{Ker }q= E\), or if \(p\) and \(q\) are similar and \(\text{Ker }p\oplus \text{Ker }q= E\). Further conditions of this type are given to imply degree 6 and 9.