On the property IR of Friis and Rordam

TL;DR

This paper investigates the IR property in C*-algebras, extending Lin's approximate commutation results, and explores its implications for algebra extensions and K-theory, revealing new cancellation properties.

Contribution

It provides a detailed study of the IR property, including its behavior under extensions and its connection to projection cancellation in C*-algebras.

Findings

IR implies a projection cancellation property between strong and weak forms.

The behavior of IR under algebra extensions is characterized.

Stable IR relates to non-stable K-theory and cancellation properties.

Abstract

In a 1997 paper Lin solved a longstanding problem as follows: For each epsilon > 0, there is delta > 0 such that if h and k are self-adjoint contractive n x n matrices and ||hk - kh|| < delta, then there are commuting self-adjoint matrices h' and k' such that ||h' - h||, ||k' - k|| < epsilon. Here delta depends only on epsilon and not on n. In a 1996 paper Friis and Rordam greatly simplified Lin's proof by using a property they called IR. They also generalized Lin's result by showing that the matrix algebras can be replaced by any C*-algebras satisfying IR. The purpose of this paper is to study the property IR. One of our results shows how IR behaves for C*-algebra extensions. Other results concern non-stable K-theory. One shows that IR (at least the stable version) implies a cancellation property for projections which is intermediate between the strong cancellation property satisfied by C*-algebras of stable rank one and the weak cancellation defined in a 2014 paper of Pedersen and the author.