Exploring new solutions to Tingley’s problem for function algebras

In this note we present two new positive answers to Tingley's problem in certain subspaces of function algebras. In the first result we prove that every surjective isometry between the unit spheres, S(A) and S(B), of two uniformly closed function algebras A and B on locally compact Hausdorff spaces can be extended to a surjective real linear isometry from A onto B. In a second part we study surjective isometrics between the unit spheres of two abelian JB*-triples represented as spaces of continuous functions of the form C-0(T)(X) := { a is an element of C-0(X) : a(lambda t) = lambda a(t) for every (lambda,t) is an element of T x X}, where X is a (locally compact Hausdorff) principal T-bundle and T denotes the unit sphere of C. We establish that every surjective isometry Delta : S(C-0(T) (X)) -> -S(C-0(T)(Y)) admits an extension to a surjective real linear isometry between these two abelian JB*-triples.