Topological reflexivity of isometries on algebras of \(C(Y)\)-valued Lipschitz maps (Q2236020)

Let \(X\) be a compact metric space, \(Y\) a compact Hausdorff space such that for the space of continuous functions, \(C(Y)\), the group of isometries is topologically reflexive. Consider the space of \(C(Y)\)-valued Lipschitz functions on \(X\), equipped with the sum norm \(\|f\|_{\infty}+\mathrm{Lip}(f)\). Under the assumption of algebraic reflexivity of \(C(Y)\), it was proved by \textit{S. Oi} [Linear Algebra Appl. 566, 167--182 (2019; Zbl 1420.46009)] that the isometry group of the vector-valued function space is also algebraically reflexive. Motivated by this and some other recent work of \textit{O. Hatori} and \textit{S. Oi} [J. Math. Anal. Appl. 452, No. 1, 378--387 (2017; Zbl 1482.47075)], \textit{S. Izumi} and \textit{H. Takagi} [J. Math. Anal. Appl. 467, No. 1, 315--330 (2018; Zbl 1401.46034)], the authors establish the topological reflexivity of the isometry group of this space of \(C(Y)\)-valued Lipschitz functions.