Projections in the convex hull of three surjective isometries on \(C(\Omega )\) (Q531897)

Let \(X\) be a complex Banach space and \(T\) be the unit circle. A projection \(P\) on \(X\) is said to be a generalized bi-circular projection (GBP) if there exists \(\lambda\in T\;\{1\}\) such that \(P+\lambda (I-P)\) is surjective isometry on \(X\). Further, \(P\) on \(X\) is said to be a generalized \(n\)-circular projection if there exists a surjective isometry \(L\) on \(X\) such that \(L^n=I\) and \(P=\frac{I+L+L^2+\dots +L^n}{n}\). In this paper, the authors give another definition of generalized \(n\)-circular projections similar to the definition of GBP (see Definition 1.1). They prove that the two definitions are equivalent. When \(n=3\), they also prove that a projection which can be written as a convex hull of three isometries is either a GBP or a generalized 3-circular projection.