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revised June 1999 to appear in Pacific J. Math.
 

Summary: revised June 1999
to appear in Pacific J. Math.
K 1 OF SEPARATIVE EXCHANGE RINGS AND
C*­ALGEBRAS WITH REAL RANK ZERO
P. Ara, K. R. Goodearl, K. C. O'Meara, and R. Raphael
Abstract. For any (unital) exchange ring R whose finitely generated projective modules
satisfy the separative cancellation property (A \Phi A ¸ = A \Phi B ¸ = B \Phi 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.
Introduction
The extent to which matrices over a ring R can be diagonalized is a measure of the
complexity of R, as well as a source of computational information about R and its free
modules. Two natural properties offer themselves as ``best possible'': (1) that an arbitrary
matrix can be reduced to a diagonal matrix on left and right multiplication by suitable
invertible matrices, or (2) that an arbitrary invertible matrix can be reduced to a diagonal
one by suitable elementary row and column operations. The second property has an

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics