A survey on Tingley's problem for operator algebras

TL;DR

This survey reviews recent advances in Tingley's problem, focusing on isometries between unit sphere subsets in various operator algebra contexts, highlighting key developments and open questions.

Contribution

It compiles and discusses the latest results on isometry extensions in operator algebras related to Tingley's problem, emphasizing new insights and unresolved issues.

Findings

Recent solutions for specific classes of operator algebras

Identification of conditions for isometry extensions

Open problems in the general case

Abstract

We survey the most recent results on extension of isometries between special subsets of the unit spheres of C$^*$-algebras, von Neumann algebras, trace class operators, preduals of von Neumann algebras, and $p$-Schatten-von Neumann spaces, with special interest on Tingley's problem.