A projection \(P\) on a complex Banach space is called bi-circular if \(e^{i\alpha} P + e^{i\beta}(I-P)\) is an isometry for all real numbers \(\alpha\) and \(\beta\), and \(P\) is a generalized bi-circular projection if \(P + \lambda (1-P)\) is an isometry for some uni-modular scalar \(\lambda \neq 1\). Let \(K\) be a compact and connected Hausdorff space. It is a result of the first-named author [F. Botelho, J. Math. Anal. Appl. 341, No. 2, 1163–1169 (2008; Zbl 1139.47026)] that the average of two isometries on \(C(K)\) is a projection if and only if it is a generalized bi-circular projection.
The object of study in this paper is projections in the convex hull of \(n\) surjective linear isometries. It is shown that, in general, if \(I_1\) and \(I_2\) are surjective isometries on a Banach space \(X\) and a non-trivial convex combination \(\lambda I_1 + (1-\lambda) I_2\) is a non-trivial projection, then \(\lambda = 1/2\).
It is shown that there exists a single projection in the convex hull of all isometries in a cyclic group of finite order, and that this projection is the average of all the isometries in the group. These results are then applied in the setting of vector-valued absolutely continuous functions on a compact subset of \(\mathbb{R}\) with at least two points, with values in a strictly convex Banach space.