A properly infinite C*-algebra which is not K_1-injective

TL;DR

This paper constructs a specific unital properly infinite C*-algebra that demonstrates it is not K_1-injective, challenging assumptions about the relationship between proper infiniteness and K_1-injectivity.

Contribution

It provides the first explicit example of a unital properly infinite C*-algebra lacking K_1-injectivity, advancing understanding of their structural properties.

Findings

Constructed a unital properly infinite C*-algebra that is not K_1-injective

Shows that proper infiniteness does not imply K_1-injectivity in C*-algebras

Provides insight into the relationship between algebraic properties and K-theory

Abstract

We construct in this note a unital properly infinite C*-algebra which is not K$_1$-injective.