### 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.

Original language | English |
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Pages (from-to) | 261-275 |

Journal | Pacific Journal of Mathematics |

Volume | 195 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Jan 2000 |

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## Cite this

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