Algebraic reflexivity of some subsets of the isometry group

Let X be a compact first countable space. in this paper we show that the set of isometrics of C(X) that are involutions is algebraically reflexive. As a consequence of a recent work of Botelho and Jamison this leads to the conclusion that the set of generalized bi-circular projections on C(X) is also algebraically reflexive. We also consider these questions for the space C (X, E) where E is a uniformly convex Banach space. (C) 2008 Elsevier Inc. All rights reserved.