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.