$K_1$ of separative exchange rings and C*-algebras with real rank zero

TL;DR

This paper proves that in separative exchange rings, invertible matrices can be diagonalized, leading to a natural isomorphism between topological K1 groups and unitary groups in certain C*-algebras with real rank zero, confirming a conjecture.

Contribution

It establishes the surjectivity of the natural homomorphism from GL_1(R) to K_1(R) for separative exchange rings and confirms a conjecture relating K_1 and unitary groups in separative C*-algebras with real rank zero.

Findings

Invertible matrices over separative exchange rings can be diagonalized.

The homomorphism from GL_1(R) to K_1(R) is surjective.

K_1 of certain C*-algebras is isomorphic to the unitary group modulo connected component.

Abstract

For any (unital) exchange ring $R$ whose finitely generated projective modules satisfy the separative cancellation property ($A\oplus A\cong A\oplus B\cong B\oplus B$ implies $A\cong B$), it is shown that all invertible square matrices over $R$ can be diagonalized by elementary row and column operations. Consequently, the natural homomorphism $GL_1(R) \to K_1(R)$ 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 $K_1(A)$ 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.