Completeness of $n$--tuple of projections in $C^*$--algebras

Let $(P_1,...,P_n)$ be an $n$--tuple of projections in a unital $C^*$--algebra $\aa$. We say $\pn$ is complete in $\aa$ if $\aa$ is the linear direct sum of the closed subspaces $P_1\aa,...,P_n\aa$. In this paper, we give some necessary and sufficient conditions for the completeness of $\pn$ and discuss the perturbation problem and topology of the set of all complete $n$--tuple of projections in $\aa$. Some interesting and significant results are obtained in this paper.