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. the first result prove that every surjective isometry between unit spheres, S(A) and S(B), uniformly closed algebras A B on locally compact Hausdor? spaces can be extended a real linear from onto B. second part study isometries spheres abelian JB*-triples represented as continuous functions formwhere X is (locally Hausdor?) principal -bundle denotes sphere ?. We establish admits an extension these JB*-triples.