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

http://dx.doi.org/10.4153/CJM-2011-092-x
Canad. J. Math. 64(2012), 755-780

  Published:2011-12-23
 Printed: Aug 2012

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.