K1-injectivity of unital properly infinite C*-algebras

Establish whether every unital properly infinite C*-algebra is $K_1$-injective.

Background

The paper’s classification theorem assumes that the corona algebra $C(B)$ is $K_1$-injective and properly infinite. The authors then list several conditions on the nonunital $\sigma$-unital C*-algebra $B$—including stability, simplicity with continuous scale, simplicity with purity and quasicontinuous scale, and $O_{\infty}$-stability—that guarantee $K_1$-injectivity of $C(B)$.

The authors explain that the remaining general question is whether proper infiniteness alone implies $K_1$-injectivity in the unital case. Resolving it would clarify whether the $K_1$-injectivity hypothesis used in the classification framework follows automatically from proper infiniteness.