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Homotopy Classification of Projections in the Corona Algebra of a Non-simple $C^*$-algebra

  Published:2011-12-23
 Printed: Aug 2012
  • Lawrence G. Brown,
    Department of Mathematics, Purdue University, West Lafayette, USA 47907
  • Hyun Ho Lee,
    Department of Mathematics, University of Ulsan, Ulsan, Korea 680-749
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Abstract

We study projections in the corona algebra of $C(X)\otimes K$, where K is the $C^*$-algebra of compact operators on a separable infinite dimensional Hilbert space and $X=[0,1],[0,\infty),(-\infty,\infty)$, or $[0,1]/\{ 0,1 \}$. Using BDF's essential codimension, we determine conditions for a projection in the corona algebra to be liftable to a projection in the multiplier algebra. We also determine the conditions for two projections to be equal in $K_0$, Murray-von Neumann equivalent, unitarily equivalent, or homotopic. In light of these characterizations, we construct examples showing that the equivalence notions above are all distinct.
Keywords: essential codimension, continuous field of Hilbert spaces, Corona algebra essential codimension, continuous field of Hilbert spaces, Corona algebra
MSC Classifications: 46L05, 46L80 show english descriptions General theory of $C^*$-algebras
$K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
46L05 - General theory of $C^*$-algebras
46L80 - $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
 

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