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

P. Ara·K. R. Goodearl·K. C. O'Meara·R. Raphael
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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.

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P. Ara, K. R. Goodearl, K. C. O'Meara, R. Raphael. 1999-06-21. $K_1$ of separative exchange rings and C*-algebras with real rank zero. https://arxiv.org/abs/math/9906141

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