Homotopies of constant Cuntz class

TL;DR

This paper proves that in certain well-behaved C*-algebras, the set of positive elements with a fixed Cuntz class forms a path-connected space, extending to irrational rotation and AF algebras.

Contribution

It establishes the path-connectedness of positive elements with fixed Cuntz class in a broad class of C*-algebras, including irrational rotation and AF algebras.

Findings

Positive elements with fixed Cuntz class are path connected in specified C*-algebras.

Applicable to irrational rotation algebras and AF algebras.

Enhances understanding of the structure of positive elements in C*-algebras.

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 Cuntz class is path connected. This result applies in particular to irrational rotation algebras and AF algebras.