Deformation of a projection in the multipleir algebra and projection lifting from the corona algebra of a non-simple C*-algebra

Let $X$ be a unit interval or a unit circle and let $B$ be a $\sigma_p$-unital, purely infinite, simple $C\sp*$-algebra such that its multiplier algebra $M(B)$ has real rank zero. Then we determine necessary and sufficient conditions for a projection in the corona algebra of $C(X)\otimes B$ to be liftable to a projection in the multiplier algebra. This generalizes a result proved by L. Brown and the author \cite{BL}. The main technical tools are divided into two parts. The first part is borrowed from the author's previous paper(JFA 260 (2011)). The second part is a proposition showing that we can produce a sub-projection, with an arbitrary rank which is prescribed as K-theoretical data, of a projection or a co-projection in the multiplier algebra of $C(X)\otimes B$ under a suitable "infinite rank and co-rank" condition.