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Homotopy Classification of Projections in the Corona Algebra of a Nonsimple $C^*$algebra


Published:20111223
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 680749
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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 $K_0$, Murrayvon
Neumann equivalent, unitarily equivalent, or homotopic. In light of
these characterizations, we construct examples showing that the
equivalence notions above are all distinct.