# Tingley's problem for spaces of trace class operators

## TL;DR

This paper solves Tingley's problem for trace class operator spaces by showing that every surjective isometry on their unit spheres extends uniquely to a linear or conjugate linear isometry of the entire space.

## Contribution

It provides the first positive solution to Tingley's problem for a new class of operator algebras, specifically trace class spaces.

## Key findings

- Every surjective isometry between unit spheres extends uniquely to a linear or conjugate linear isometry.
- The result applies to trace class spaces, expanding the classes where Tingley's problem is solved.
- This advances understanding of the geometric structure of operator algebras.

## Abstract

We prove that every surjective isometry between the unit spheres of two trace class spaces admits a unique extension to a surjective complex linear or conjugate linear isometry between the spaces. This provides a positive solution to Tingley's problem in a new class of operator algebras.