The group structure of the homotopy set whose target is the automorphism group of the Cuntz algebra

TL;DR

This paper characterizes the group structure of the homotopy set targeting automorphisms of the Cuntz algebra using K-theory, revealing non-commutative examples and refining classification results.

Contribution

It determines the homotopy set's group structure for automorphisms of Cuntz algebras and improves classification of continuous fields via vector bundles.

Findings

Homotopy set forms a non-commutative group in some cases

Classifying space of automorphisms is not an H-space

Enhanced classification of continuous fields of Cuntz algebras

Abstract

We determine the group structure of the homotopy set whose target is the automorphism group of the Cuntz algebra $O_{n+1}$ for finite n in terms of K-theory. We show that there is an example of a space for which the homotopy set is a non-commutative group, and hence the classifying space of the automorphism group of the Cuntz algebra for finite n is not an H-space. We also make an improvement of Dadarlat's classification of continuous fields of the Cuntz algebras in terms of vector bundles.