1999/06/21 by P. Ara, Ara, P., K. R. Goodearl +5
Mathematics · #15A33 #16E50 #19B14 #46L80 #FOS: Mathematics #Operator Algebras (math.OA) #Rings and Algebras (math.RA) #math.OA #math.RA #msc:15A33 #msc:16E50 #msc:19B14 #msc:46L80
paper · pdf · doi:10.48550/arxiv.math/9906141
12 pages; to appear in Pacific J. Math
arxiv created 1999/06/21 · arxiv updated 2009/11/30
For any (unital) exchange ring R whose finitely generated projective modules satisfy the separative cancellation property (A⊕ A≅ A⊕ B≅ B⊕ B implies 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.