Higher homotopy groups of Cuntz classes

TL;DR

This paper computes the homotopy groups of Cuntz classes in certain C*-algebras, extending previous results and covering examples like AF algebras and noncommutative tori.

Contribution

It provides a complete calculation of the homotopy groups of Cuntz classes for a broad class of C*-algebras, including non-compact classes.

Findings

Homotopy groups of positive elements with non-compact Cuntz class vanish.

Complete homotopy group calculations for these classes.

Includes examples such as AF algebras and irrational noncommutative tori.

Abstract

Let $A$ be a unital simple separable exact C$^*$-algebra which is approximately divisible and of real rank zero. We prove that the set of positive elements in $A$ with a fixed non-compact Cuntz class has vanishing homotopy groups. Combined with work of S. Zhang for the case of compact elements, this gives a complete calculation of the homotopy groups of Cuntz classes for these algebras. Examples covered include approximately finite-dimensional (AF) algebras and irrational noncommutative tori.