Behaviour of holomorphic automorphisms on equicontinuous subsets of the space C(Omega, E)

Consider a compact Hausdorff topological space Omega, a JB*-triple E and F := C(Omega, E), the JB*-triple of all continuous E-valued functions f: Omeg a --> E with the pointwise operations and the norm of the supremum. Let G b e the group of all holomorphic automorphisms of the unit ball B-F of F that map every equicontinuous subset lying strictly inside B-F into another suc h a set. The real Banach-Lie group G and its Lie algebra are investigated. The identity connected component of G is identified when E has the strong B anach-Stone property. This extends to the infinite dimensional setting a we ll known result concerning the case E = C.