Approximate Unitary Equivalence of Homomorphisms from O_infinity

TL;DR

This paper proves that homomorphisms from the Cuntz algebra O_infinity 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 of certain homomorphisms from O_infinity, extending Rordam's classification theorem to more general direct limits involving O_infinity.

Findings

Homomorphisms with same KK-class are approximately unitarily equivalent.

Classification extends to direct limits involving corners of O_infinity.

Results apply to K_0 groups with arbitrary countable abelian groups without even torsion.

Abstract

We prove that if two nonzero homomorphisms from the Cuntz algebra O_infinity 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_infinity 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_infinity for which the K_0 group can be an arbitrary countable abelian group with no even torsion.