## Annals of Functional Analysis

- Ann. Funct. Anal.
- Volume 8, Number 4 (2017), 460-472.

### Nonlinear isometries between function spaces

Kathleen Roberts and Kristopher Lee

#### Abstract

We demonstrate that any surjective isometry $T:\mathcal{A}\to \mathcal{B}$ not assumed to be linear between unital, completely regular subspaces of complex-valued, continuous functions on compact Hausdorff spaces is of the form $$T\left(f\right)=T\left(0\right)+Re[\mu \cdot (f\circ \tau \left)\right]+iIm[\nu \cdot (f\circ \rho \left)\right],$$ where $\mu $ and $\nu $ are continuous and unimodular, there exists a clopen set $K$ with $\nu =\mu $ on $K$ and $\nu =-\mu $ on ${K}^{c}$, and $\tau $ and $\rho $ are homeomorphisms.

#### Article information

**Source**

Ann. Funct. Anal., Volume 8, Number 4 (2017), 460-472.

**Dates**

Received: 16 July 2016

Accepted: 13 December 2016

First available in Project Euclid: 2 June 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.afa/1496368961

**Digital Object Identifier**

doi:10.1215/20088752-2017-0010

**Mathematical Reviews number (MathSciNet)**

MR3717168

**Zentralblatt MATH identifier**

06841327

**Subjects**

Primary: 46B04: Isometric theory of Banach spaces

Secondary: 46E25: Rings and algebras of continuous, differentiable or analytic functions {For Banach function algebras, see 46J10, 46J15}

**Keywords**

isometry nonlinear function spaces

#### Citation

Roberts, Kathleen; Lee, Kristopher. Nonlinear isometries between function spaces. Ann. Funct. Anal. 8 (2017), no. 4, 460--472. doi:10.1215/20088752-2017-0010. https://projecteuclid.org/euclid.afa/1496368961