For any (unital) exchange ring R whose finitely generated projective modules satisfy the separative cancellation property (A ⊕ A ≅ A ⊕ B ≅ B ⊕ B ⇒ A ≅ B), it is shown that all invertible square matrices over R can be diagonalized by elementary row and column operations. Consequently, the natural homomorphism GL1(R) → K1(R) is surjective. In combination with a result of Huaxin Lin, it follows that for any separative, unital C*-algebra A with real rank zero, the topological K1(A) is naturally isomorphic to the unitary group U(A) modulo the connected component of the identity. This verifies, in the separative case, a conjecture of Shuang Zhang.
Ara, P., Goodearl, K. R., O'Meara, K. C., & Raphael, R. (2000). K<inf>1</inf> of separative exchange rings and C*-algebras with real rank zero. Pacific Journal of Mathematics, 195(2), 261-275. https://doi.org/10.2140/pjm.2000.195.261