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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
 MSC Classifications: 46L05 - General theory of $C^*$-algebras 46L80 - $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]