Contractible Cuntz classes in $Z$-stable C$^*$-algebras

TL;DR

This paper proves that certain positive elements in Z-stable C*-algebras form contractible spaces and computes their homotopy groups, advancing understanding of their topological structure.

Contribution

It establishes the contractibility of fixed non-compact Cuntz classes in Z-stable C*-algebras and provides a full calculation of their homotopy groups.

Findings

Positive elements in C*-algebras form contractible spaces

Complete calculation of homotopy groups of Cuntz classes

Extension of previous results to non-compact classes

Abstract

Let $A$ be a unital, simple and Z-stable C$^*$-algebra. We show that the set of positive elements in $A$ (resp. $A \otimes K$) belonging to a fixed non-compact Cuntz class is contractible as a topological subspace of $A$ (resp. $A \otimes K$). In light of earlier work by Zhang, Jiang and Hua in the compact case, we deduce a complete calculation of the homotopy groups of Cuntz classes for these algebras.