A note on lifting projections

‎Suppose $\pi:\mathcal{A}\rightarrow \mathcal{B}$ is a surjective unital $\ast‎ ‎$-homomorphism between C*-algebras $\mathcal{A}$ and $\mathcal{B}$‎, ‎and $0\leq‎ ‎a\leq1$ with $a\in \mathcal{A}$‎. ‎We give a sufficient condition that ensures‎ ‎there is a proection $p\in \mathcal{A}$ such that $\pi \left( p\right)‎ ‎=\pi \left( a\right) $‎. ‎An easy consequence is a result of [L‎. ‎G‎. ‎Brown and G‎. ‎k‎. ‎Pedersen‎, ‎C*-algebras of real rank‎ ‎zero‎, ‎\textit{J‎. ‎Funct‎. ‎Anal.} {99} (1991) 131--149] that such a $p$ exists when $\mathcal{A}$ has real rank zero‎.