Surjective isometries on the Banach algebra of continuously differentiable maps with values in Lipschitz algebra

Abstract

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