Deformation of a projection in the multiplier algebra and projection lifting from the corona algebra of a non-simple C⁎C⁎-algebra

Let X be a unit interval or a unit circle and let B   be a σpσp-unital, purely infinite, simple C⁎C⁎-algebra such that its multiplier algebra M(B)M(B) has real rank zero. Then we determine necessary and sufficient conditions for a projection in the corona algebra of C(X)⊗BC(X)⊗B to be liftable to a projection in the multiplier algebra. This generalizes a result proved by L. Brown and the author, Brown and Lee (2012) [3]. The main technical tools are divided into two parts. The first part is borrowed from the authorʼs result, Lee (2011) [14, Theorem 3.3]. 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)⊗BC(X)⊗B under a suitable “infinite rank and co-rank” condition.