Weakening idempotency in $K$-theory

Vladimir Manuilov

Vol. 289 (2017), No. 2, 381–394

DOI: 10.2140/pjm.2017.289.381

Abstract

We show that the $K$-theory of $C^{\ast }$-algebras can be defined by pairs of matrices $a,b$ satisfying less strict relations than idempotency, namely $p\left(a\right)=p\left(b\right)$ for any polynomial $p$ with $p\left(0\right)=p\left(1\right)=0$, which means that $a$ and $b$ have to be “projections” only where $a\ne b$.

Keywords

$K$-theory, $C^*$-algebra, projection

Mathematical Subject Classification 2010

Primary: 19K99, 46L80, 46L05

Milestones

Received: 10 November 2015

Revised: 20 August 2016

Accepted: 30 December 2016

Published: 19 June 2017