Tingley’s problem for the direct sum of uniformly closed extremely C-regular subspaces with the \(\ell ^{1}\)-sum norm

Abstract

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 \(\{A_\mu \}_{\mu \in M}\) and \(\{A_{\nu }\}_{\nu \in N}\) be two collections of uniformly closed extremely C-regular subspaces. In this paper, we prove that if \(\Delta \) is a surjective isometry between two unit spheres of \(\ell ^1\)-sums of uniformly closed extremely C-regular subspaces \(\{A_{\mu }\}_{\mu \in M}\) and \(\{A_{\nu }\}_{\nu \in N}\), then \(\Delta \) 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 \(\Vert f\Vert _{1}=|f(0)|+\Vert f'\Vert _{\infty }\) for \(f\in C^1(I)\), \(C^{(n)}(I)\) of all n-times continuously differentiable complex-valued functions on I with the norm \(\Vert f\Vert _{1}=\sum _{k=0}^{n-1}|f^{(k)}(0)|+~\Vert f^{(n)}\Vert _{\infty }\) for \(C^{n}(I)\), and \(\ell ^1(\mathbb {N})\) of all complex-valued functions on the set \(\mathbb {N}\) of all natural numbers with the norm \(\Vert a\Vert _{1}=\sum _{n\in \mathbb {N}}|a(n)|\) for \(a\in \ell ^1(\mathbb {N})\).