Abstract

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.

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.

For any (unital) exchange ring $R$ whose finitely generated projective modules satisfy the separative cancellation property ($A\oplus A\cong A\oplus B\cong B\oplus B\Rightarrow A\cong B$), it is shown that all invertible square matrices over $R$ can be diagonalized by elementary row and column operations. Consequently, the natural homomorphism $G{L}_1\left(R\right)\to K_1\left(R\right)$ 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 $K_1\left(A\right)$ is naturally isomorphic to the unitary group $U\left(A\right)$ modulo the connected component of the identity. This verifies, in the separative case, a conjecture of Shuang Zhang.

For any (unital) exchange ring $R$ whose finitely generated projective modules satisfy the separative cancellation property ($A\oplus A\cong A\oplus B\cong B\oplus B\Rightarrow A\cong B$), it is shown that all invertible square matrices over $R$ can be diagonalized by elementary row and column operations. Consequently, the natural homomorphism $G{L}_1\left(R\right)\to K_1\left(R\right)$ 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 $K_1\left(A\right)$ is naturally isomorphic to the unitary group $U\left(A\right)$ modulo the connected component of the identity. This verifies, in the separative case, a conjecture of Shuang Zhang.

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.

Authors