$K_1$-injectivity of the Paschke dual algebra, and uniqueness

TL;DR

This paper proves that many Paschke dual algebras of simple unital C*-algebras are $K_1$-injective, leading to new $KK$-uniqueness theorems that extend classical essential codimension results.

Contribution

It establishes $K_1$-injectivity for a broad class of Paschke dual algebras and derives generalized $KK$-uniqueness theorems.

Findings

Many Paschke dual algebras are $K_1$-injective

Derived new $KK$-uniqueness theorems

Extended the Brown--Douglas--Fillmore essential codimension property

Abstract

We prove that a large class of Paschke dual algebras of simple unital C*-algebras are $K_1$-injective. As a consequence, we obtain interesting $KK$-uniqueness theorems which generalize the Brown--Douglas--Fillmore essential codimension property.