Exploring new solutions to Tingley's problem for function algebras

Abstract

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 goal we study surjective isometries between the unit spheres of two abelian JB * -triples represented as spaces of continuous functions of the formfor every (λ, t) ∈ T × X}, where X is a (locally compact Hausdorff) principal T-bundle. We establish that every surjective isometry ∆ : S(C T 0 (X)) → S(C T 0 (Y )) admits an extension to a surjective real linear isometry between these two abelian JB * -triples.

Every commutative JB$^*$-triple satisfies the complex Mazur--Ulam property

“…for every (λ, t) ∈ T × L}, where L is a principal T-bundle L, that is, a subset of a Hausdorff locally convex complex space such that 0 / ∈ L, L ∪ {0} is compact, and TL = L (see also [8]). We can state next the main result of the paper.…”

“…The next remark, which has been borrowed from [8,Remark 3.4] states a kind of Urysohn's lemma for the space C T 0 (L). We can now begin with the technical details for our arguments.…”

“…The non-unital analogue of uniform algebras is materialized in the notion of uniformly closed function algebra on a locally compact Hausdorff space L. We recently showed that each surjective isometry between the unit spheres of two uniformly closed function algebras on locally compact Hausdorff spaces admits an extension to a surjective real linear isometry between these algebras (see [8]). In the just quoted reference we also proved that Tingley's problem admits a positive solution for any surjective isometry between the unit spheres of two commutative JB * -triples, which are not, in general, subalgebras of the algebra C 0 (L) of all complex-valued continuous functions on L vanishing at infinity (see Section 3 for the detailed representation of commutative JB * -triples).…”