On the property IR of Friis and Rørdam

Abstract

Lin solved a longstanding problem as follows. For each $ϵ>0$, there is $\delta >0$ such that, if $h$ and $k$ are self-adjoint contractive $n\times n$ matrices and $\Vert hk-kh\Vert <\delta$, then there are commuting self-adjoint matrices $h\text{'}$ and $k\text{'}$ such that $\Vert h\text{'}-h\Vert$, $\Vert k\text{'}-k\Vert <ϵ$. Here $\delta$ depends only on $ϵ$ and not on $n$. Friis and Rørdam greatly simplified Lin’s proof by using a property they called $\mathit{IR}$. They also generalized Lin’s result by showing that the matrix algebras can be replaced by any $C^{\ast }$-algebras satisfying $\mathit{IR}$. The purpose of this paper is to study the property $\mathit{IR}$. One of our results shows how $\mathit{IR}$ behaves for $C^{\ast }$-algebra extensions. Other results concern nonstable $K$-theory. One shows that $\mathit{IR}$ (at least the stable version) implies a cancellation property for projections which is intermediate between the strong cancellation satisfied by $C^{\ast }$-algebras of stable rank $1$ and the weak cancellation defined in a 2014 paper by Pedersen and the author.