Completeness of $n$--tuple of projections in $C^{\ast }$--algebras

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