Tingley's problem through the facial structure of operator algebras

TL;DR

This paper affirms Tingley's problem for specific classes of operator algebra spaces, demonstrating that surjective isometries on unit spheres extend to linear isometries and relate to unitary groups.

Contribution

It provides the first comprehensive solution to Tingley's problem for preduals, self-adjoint operators, and normal functionals of von Neumann algebras, and explores isometries between state spaces.

Findings

Surjective isometries extend to linear isometries in specified operator algebra spaces.

Isometries between unit spheres induce bijections between unitary groups.

Normal state space isometries extend to linear isometries.

Abstract

Tingley's problem asks whether every surjective isometry between the unit spheres of two Banach spaces admits an extension to a real linear surjective isometry between the whole spaces. In this paper, we give an affirmative answer to Tingley's problem when both spaces are preduals of von Neumann algebras, the spaces of self-adjoint operators in von Neumann algebras or the spaces of self-adjoint normal functionals on von Neumann algebras. We also show that every surjective isometry between the unit spheres of unital C$^*$-algebras restricts to a bijection between their unitary groups. In addition, we show that every surjective isometry between the normal state spaces or the normal quasi-state spaces of two von Neumann algebras extends to a linear surjective isometry.