arXiv math/99061411999-06-21math.RAmath.OA

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

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

备注：: 12 pages; to appear in Pacific J. Math

英文摘要：

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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