Proper asymptotic unitary equivalence in \KK-theory and projection lifting from the corona algebra

In this paper we generalize the notion of essential codimension of Brown, Douglas, and Fillmore using \KK-theory and prove a result which asserts that there is a unitary of the form `identity + compact' which gives the unitary equivalence of two projections if the `essential codimension' of two projections vanishes for certain C\sp*-algebras employing the proper asymptotic unitary equivalence of \KK-theory found by M. Dadarlat and S. Eilers. We also apply our result to study the projections in the corona algebra of $C(X)\otimes B$ where $X$ is $[0,1]$, $(-\infty, \infty)$, $[0,\infty)$, and $[0,1]/\{0,1\}$.Comment: 14 pages, to appear in JF