K_1-injectivity for properly infinite C*-algebras

TL;DR

This paper investigates the K_1-injectivity property in properly infinite C*-algebras, building on Cuntz's results for purely infinite simple C*-algebras, and explores its implications for classification.

Contribution

It introduces a connection between the K_1-injectivity of a finitely generated C*-algebra and all unital properly infinite C*-algebras, advancing classification techniques.

Findings

K_1-injectivity of a finitely generated algebra implies it for all properly infinite C*-algebras

Extension of Cuntz's results to broader classes of C*-algebras

Potential simplification in classifying properly infinite C*-algebras

Abstract

One of the main tools to classify \cst-algebras is the study of its projections and its unitaries. It was proved by Cuntz in \cite{Cu81} that if $A$ is a \textit{purely infinite} simple \cst-algebra, then the kernel of the natural map for the unitary group $\U(A)$ to the $K$-theory group $K_1(A)$ is reduced to the connected component $\U^0(A)$, i.e. $A$ is \textit{$K_1$-injective} (see \S 3). We study in this note a finitely generated \cst-algebra, the $K_1$-injectivity of which would imply the $K_1$-injectivity of all unital \textit{properly infinite} \cst-algebras.