Tingley’s problem asks whether every surjective isometry between two unit spheres of Banach spaces can be extended to a surjective real linear isometry between the whole spaces. Let {A_μ}_μ∈ M and {A_ν}_ν∈ N be two collections of uniformly closed extremely C-regular subspaces. In this paper, we prove that if Δ is a surjective isometry between two unit spheres of ℓ ^1 -sums of uniformly closed extremely C-regular subspaces {A_μ}_μ∈ M and {A_ν}_ν∈ N , then Δ admits an extension to a surjective real linear isometry between the whole spaces. Typical examples of such Banach spaces B are C^1(I) of all continuously differentiable complex-valued functions on the closed unit interval I equipped with the norm ‖ f‖ _1=|f(0)|+‖ f'‖ _∞ for f∈ C^1(I) , C^(n)(I) of all n-times continuously differentiable complex-valued functions on I with the norm ‖ f‖ _1=∑ _k=0^n-1|f^(k)(0)|+ ‖ f^(n)‖ _∞ for C^n(I) , and ℓ ^1(ℕ) of all complex-valued functions on the set ℕ of all natural numbers with the norm ‖ a‖ _1=∑ _n∈ℕ|a(n)| for a∈ℓ ^1(ℕ) .

For a locally compact Hausdorff space L, we denote by C_0(L,ℝ) the Banach space of all continuous real-valued functions on L vanishing at infinity equipped with the supremum norm. We prove that every surjective phase-isometry T:C_0^+(X,ℝ)→ C_0^+(Y,ℝ) between the positive cones of C_0(X,ℝ) and C_0(Y,ℝ) is a composition operator induced by a homeomorphism between X and Y. Furthermore, we show that any surjective phase-isometry T:C_0^+(X,ℝ)→ C_0^+(Y,ℝ) extends to a surjective linear isometry from C_0(X,ℝ) onto C_0(Y,ℝ) .

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.

Let Lip(I) be the Banach algebra of all Lipschitz functions on the closed unit interval I with the norm ‖ f‖ _L=‖ f‖ _∞ +L(f) for f∈Lip(I) , where L ( f ) is the Lipschitz constant of f . We denote by C^1(I, Lip(I)) the Banach algebra of all continuously differentiable functions F from I to Lip(I) equipped with the norm ‖ F‖ _Σ=sup _s∈ I‖ F(s)‖ _L+sup _t∈ I‖ D(F)(t)‖ _L for F∈ C^1(I, Lip(I)) . In this paper, we prove that if T is a surjective, not necessarily linear, isometry on C^1(I, Lip(I)) , then T-T(0) is a weighted composition operator or its complex conjugation. Among other things, any surjective complex linear isometry on C^1(I, Lip(I)) is of the following form: c_1F(τ _1(s),τ _2(x)) , where c_1 is a complex number of modulus 1, and τ _1 and τ _2 are isometries of I onto itself.

We prove that every commutative JB $$^*$$ -triple, represented as a space of continuous functions $$C_0^{\mathbb {T}}(L),$$ satisfies the complex Mazur–Ulam property, that is, every surjective isometry from the unit sphere of $$C_0^{\mathbb {T}}(L)$$ onto the unit sphere of any complex Banach space admits an extension to a surjective real linear isometry between the spaces.

This paper is, in a first stage, devoted to establishing a topological-algebraic characterization of the principal component, U0(M), of the set of unitary elements, U(M), in a unital JB*- algebra M. We arrive to the conclusion that, as in the case of unital C*-algebras, U0(M) = M-1 1 n U(M) ? ? = Ue.nn middotmiddotmiddot Ue.n1 (1) : n E N, hj E Msa v 1 < j < n = {u E U(M) : there exists w E U0(M) with ilu -wil < 2} is analytically arcwise connected. Actually, U0(M) is the smallest quadratic subset of U(M) containing the set eiMB6. Our second goal is to provide a complete description of the surjective isometries between the principal components of two unital JB*-algebras M and N. Contrary to the case of unital C*-algebras, we shall deduce the existence of connected components in 14(M) which are not isometric as metric spaces. We shall also establish necessary and sufficient conditions to guarantee that a surjective isometry & UDelta; : 14(M)-+ 14(N) admits an extension to a surjective linear isometry between M and N, a conclusion which is not always true. Among the consequences it is proved that M and N are Jordan *- isomorphic if, and only if, their principal components are isometric as metric spaces if, and only if, there exists a surjective isometry & UDelta; : 14(M)-+ 14(N) mapping the unit of M to an element in 140(N). These results provide an extension to the setting of unital JB*-algebras of the results obtained by O. Hatori for unital C*-algebras.(c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).