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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
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.