Approximate unitary equivalence of homomorphisms from O_{\infty}

TL;DR

This paper proves that homomorphisms from the infinite Cuntz algebra O_{} to purely infinite simple C*-algebras are approximately unitarily equivalent if they share the same KK-theory class and are either both unital or both nonunital, extending classification results.

Contribution

It establishes approximate unitary equivalence for homomorphisms from O_{} based on KK-theory, broadening classification theorems to include more complex algebraic structures.

Findings

Homomorphisms with same KK-class are approximately unitarily equivalent.

Extension of Rordam's classification to direct limits involving O_{}.

K_0-groups can be any countable abelian group with no even torsion.

Abstract

We prove that if two homomorphisms from O_{\infty} to a purely infinite simple C*-algebra have the same class in KK-theory, and if either both are unital or both are nonunital, then they are approximately unitarily equivalent. It follows that O_{\infty} is classifiable in the sense of Rordam. In particular, Rordam's classification theorem for direct limits of matrix algebras over even Cuntz algebras extends to direct limits involving both matrix algebras over even Cuntz algebras and corners of O_{\infty}, for which the K_0-group can be an arbitrary countable abelian group with no even torsion.