A remark on the similarity and perturbation problems

TL;DR

This paper explores the connection between Kadison's similarity problem for C*-algebras and a perturbation theory question about whether close C*-algebras necessarily have close commutants, highlighting an equivalence between these problems.

Contribution

It establishes the equivalence between Kadison's similarity problem and a perturbation theory problem concerning the closeness of commutants of C*-algebras.

Findings

Kadison's similarity problem is equivalent to a perturbation problem.

Close C*-algebras may have non-close commutants, depending on the problem.

The paper clarifies the relationship between algebraic similarity and perturbation stability.

Abstract

In this note we show that Kadison's similarity problem for C*-algebras is equivalent to a problem in perturbation theory: must close C*-algebras have close commutants?